A cuboid has a square base of side (2 + √3)m. the area of one side is (2√3 - 3)m². find the height of the cuboid in the form (a+ b√3)m, where a and b are integers.​

3BE² + 3DF²- (2AC²- AB) is the answer of the following question

The calculation is as follows

Let E and F be the feet of the perpendiculars from B and D, respectively, to AC. We can use the fact that the area of a quadrilateral is equal to half the product of the diagonals multiplied by the sine of the angle between them to find the length of the diagonal BD.

Since ABCS is a quadrilateral, we have:

Area of ABCS = (1/2) * AC * BD * sin(angle between AC and BD)

Substituting the given values, we get:

924 = (1/2) * 33 * BD * sin(angle between AC and BD)

sin(angle between AC and BD) = 924 / (16.5 * BD)

Now, consider triangles ABC and ACD. Using the Pythagorean theorem, we can write:

AB² + BC² = AC² (1)

CD²+ BC² = AC² (2)

Adding equations (1) and (2), we get:

AB² + 2BC²+ CD²= 2AC²

Substituting AC = 33 and rearranging, we get:

BC² = (2AC²- AB² - CD²) / 2

We can also write:

BE²= AB²- AE² (3)

DF² = CD²- CF²(4)

Adding equations (3) and (4), we get:

BE² + DF²= AB²+ CD² - AE²- CF²

Substituting BC²from earlier, we get:

BE²+ DF² = 2AC² - BC²- AE²- CF²

We want to find BE + DF. Squaring both sides of equation (3), we get:

BE²= AB² - AE²

AE²= AB²- BE²

Similarly, squaring both sides of equation (4), we get:

DF²= CD²- CF²

CF²= CD² - DF²

Substituting these expressions into the equation for BE²+ DF², we get:

BE²+ DF² = 2AC² - BC² - (AB² - BE²) - (CD²- DF²)

Simplifying, we get:

BE² + DF²= 2AC² - BC²- AB²- CD² + 2BE² + 2DF²

Collecting like terms, we get:

BE²+ DF²- 2BE² - 2DF²= 2AC²- BC² - AB² - CD²

Simplifying, we get:

BE²- DF² = 2AC²- BC²- AB²- CD²- 2BE² - 2DF²

Substituting the values we know, we get:

BE²- DF²= 2(33)²- BC²- AB² - CD²- 2BE²- 2DF²

Rearranging, we get:

3BE²+ 3DF² - BC²- AB²- CD²= 2(33)² - 924

Substituting BC^2 from earlier, we get:

3BE²+ 3DF² - (2AC²- AB)

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Therefore, the prism's base can be either 9 x 9 inches, 6 x 6 inches, or

4 x 4 inches in size.

.

The amount of space that an object's surface takes up in total is measured by its surface area. It is usually expressed in square units, such as square inches or square metres.The surface area of a three-dimensional object can be determined by adding up the areas of all of its faces.

For instance, the surface area of a rectangular prism can be calculated using the formula below:

A = lw + lh + lw

The formula SA = 2lw + 2lh + 2wh, where l is the prism's length, w is its width, and h is its height, determines the surface area of a rectangular prism. Inputting the values provided yields:

306 is equal to 2l + 2(24) + 2(24) + 48

306 - 48l = 2w(l+24) 153 - 24

l = wl + 48w 153 - 48w = l(w+24) 153 - 24l - 48w = wl

We can identify all potential values of l and w that meet this equation because they are both whole numbers. Since it is a rectangular prism with a square base, we know that l > w. Given that the surface area cannot be more than 306, we also know that l(w+24) 153. We may therefore begin by identifying all variables of (153-48w) bigger than w.and equal to or less than 12 (since w cannot be greater than 12). We get:

W=1, L=9, L=2, L=6, and L=3

Therefore, the prism's base can be either 9 x 9 inches, 6 x 6 inches, or 4 x 4 inches in size.

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The value of the constant in the simplified expression are 3, - 1, and -11.

Simplification of an algebraic expression can be defined as the process of writing an expression in the most efficient and compact form without affecting the value of the original expression.

The given expression;

2(x - 6)(x + 2) + (7x + x² + 13)

Simplify the expression in the bracket as follows;

2(x² + 2x - 6x - 12) +  (7x + x² + 13)

= 2(x² -4x - 12) +  (7x + x² + 13)

Simplify the expression in the bracket by multiplying out with 2 as follows;

2(x² -4x - 12) +  (7x + x² + 13)

= 2x² - 8x - 24 + 7x + x² + 13

Collect similar terms as follows;

= 3x² - x - 11

Thus,  the value of the constant in the simplified expression are 3, - 1, and -11.

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Answer:

2n = -4

Step-by-step explanation:

This is because the N is unknown and in the way it's written it means 2 times something is -4

Answer:

\(x + 2 = - 3x\)

\( - 4x = 2\)

\(x = - \frac{1}{2} \)

\( - 3( - \frac{1}{2} ) = \frac{3}{2} = 1 \frac{1}{2} \)

So the lines intersect at (-1/2, 1 1/2), or

(-.5, 1.5).

The probability of obtaining a 6 and a head is 1/12 if Maureen rolls a fair dice and flips a coin.

It is a likeliness of happening an event. It lies between 0 to 1 .

Probability= Number of events/number of outcomes

When Maureen rolls a fair dice then probability of obtaining a six is 1/6

When Maureen flips a fair coin then probability of obtaining a head is 1/2

So the probability of obtaining a six and a head is 1/6*1/2

=1/12

Hence the probability of obtaining a six and a head is 1/12

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The two equations are equivalent and represent the same line since the second equation can be obtained from the first equation by multiplying both sides by 3.

The given equations are:2y = x + 10 ..........(1)3y = 3x + 15 .......(2)

Let us check the properties of the equations given, we get:

Properties of equation 1:It is a linear equation in two variables x and y.

It can be represented in the form y = (1/2)x + 5.

This equation is represented in the slope-intercept form where the slope (m) is 1/2 and the y-intercept (c) is 5.Properties of equation 2:

It is a linear equation in two variables x and y.

It can be represented in the form y = x + 5.

This equation is represented in the slope-intercept form where the slope (m) is 1 and the y-intercept (c) is 5.

From the above information, we can conclude that both equations are linear and have a y-intercept of 5.

However, the slope of equation 1 is 1/2 while the slope of equation 2 is 1, thus the equations have different slopes.

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Answer:

370

Step-by-step explanation:

370

1840

Step-by-step explanation:

20 * 20 + 77 - 100

Answer:

The probability that the customer finds exactly one defective light bulb among the eight purchased is approximately 0.042 or 4.2%.

Step-by-step explanation:

To find the probability that the customer finds exactly one defective light bulb among the eight they purchased, we can use the formula for combinations and probability.

1. Calculate the number of ways to choose one defective light bulb and seven non-defective light bulbs: -

Number of ways to choose 1 defective light bulb:

C(5,1) = 5! / (1! * (5-1)!) = 5

Number of ways to choose 7 non-defective light bulbs:

C(45,7) = 45! / (7! * (45-7)!) = 453,024

2. Multiply the number of ways together: -

5 (number of ways to choose 1 defective) * 453,024 (number of ways to choose 7 non-defective) = 2,265,120 (total ways to choose exactly 1 defective and 7 non-defective light bulbs)

3. Calculate the total possible ways to choose any 8 light bulbs from the 50 available: - C(50,8) = 50! / (8! * (50-8)!) = 53,907,800

4. Divide the favorable outcomes (exactly 1 defective and 7 non-defective) by the total possible outcomes: -

Probability = 2,265,120 (favorable outcomes) / 53,907,800 (total outcomes) ≈ 0.042

Therefore, the probability that the customer finds exactly one defective light bulb among the eight purchased is approximately 0.042 or 4.2%.

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John has the same preference as Kim. John loves to eat mango and the banana. Kim loves to eat mango but not the apple. Jim loves to eat banana but not the mango. John and Kim have the same preference for mango.

John's preference for mango can be inferred from the given information that he loves to eat mango and the banana. However, it does not mention anything about his preference for apples. Kim, on the other hand, loves to eat mango but not the apple. This aligns with John's preference for mango.

Among the given children, John and Kim have the same preference for fruits. They both love to eat mango, while John also enjoys eating bananas.

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A differential equation's existence and uniqueness in mathematics refers to the existence of a single, clearly defined solution that meets a certain set of requirements.

A differential equation is a type of mathematical equation that links a number of known functions or variables to an unknown function and its derivatives. If a differential equation has existence and uniqueness of solution, it means that there is only one function that satisfies the equation and corresponds to the given conditions for a given set of beginning circumstances.

The terms and structure of the differential equation, such as linearity and initial conditions, decide whether or not a solution exists. The Piccard-Lipschitz theorem, which asserts that if a function and its derivatives are locally Lipschitz continuous, then the solution to the differential equation is unique in a neighbourhood of the initial conditions, frequently ensures the uniqueness of the solution.

In conclusion, the existence and uniqueness of a solution in differential equations is a crucial idea in mathematical modelling and aids in making sure that the solutions to a given problem are clear-cut and consistent with the underlying conditions.

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Answer:

3 1/5

Step-by-step explanation:

The height of the tree is 6.4m

the difference is that in one case the shadow of the tree is 60 - 20 = 40 m, and in the other case it is 60 + 20 = 80 m long.

anyway, for a certain angle of the sunlight we know the shadow of the pole is 60 m long, which creates with the 4.8 m height a right-angled triangle, with the line of sight from the top of the pole to the ground end of the shadow being the Hypotenuse or radius for or trigonometric triangle in a circle.

the shadow length (60 m) is sine of the sunshine angle multiplied by the radius.

the height (4.8 m) is cosine of that sunshine angle multiplied also by that radius.

Pythagoras gets us the radius for the pole triangle :

radius² = 60² + 4.8² = 3600 + 23.04 = 3623.04

radius = sqrt(3623.04) = 60.19169378... m

sin(angle) × radius = 60

sin(angle) = 60/radius = 0.996815279...

angle = 85.42607874...°

the sunshine creates with the tree also a right-angled triangle with the same sunshine angle (the sun is so far away, that the tiny, tiny difference is irrelevant, we can simply assume the angles are equal).

but the shadow is of different length (sin(angle)×radius), which means also the radius for the tree triangle has to be different. and that defines the height of the tree (cos(angle)×radius).

but now we know the angle, and we can reverse calculate the side lengths.

but the question is (as mentioned at the beginning) : is the tree shadow 40 m or 80 m long.

we have therefore 2 solutions for the height of the tree.

1. the shadow is 40 m long.

sin(angle) × radius = 40

radius = 40/sin(angle) = 40.12779585... m

tree height = cos(angle)×radius = 3.2 m

2. the shadow is 80 m long.

sin(angle) × radius = 80

radius = 80/sin(angle) = 80.25559171... m

tree height = cos(angle)×radius = 6.4 m

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The equation of a line in slope-intercept form is y = mx + b, where m is the slope of the line and b is the y-intercept.

To find the slope of the line, we can use the following formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the coordinates (2,4) and (-2,-8) into the formula:

m = (-8 - 4) / (-2 - 2) = -12/ -4 = 3

We can use one of the coordinates to find the y-intercept.

Plugging in the point (2,4) and the slope 3 into the equation y = mx + b, we get:

4 = 3*2 + b

so b = -2

Therefore, the equation of the line is y = 3x - 2.

The angle between AFE is measured (A) 42°.

Since ΔABC is an equilateral triangle, all its angles are 60°. Since CAD-18°,: ∠CAE = ∠CAD + ∠DAE = 18° + 60° = 78°.

Since AC is the angle bisector of ∠BCD,:

∠ACB = ∠ACD = (1/2)∠BCD. Since ΔABC is equilateral, ∠BCA = 60°.

Therefore, ∠BCD = ∠BCA + ∠ACB = 60° + (1/2)∠BCD, which implies that ∠ACB = 30°.

Since BE- CD,:

∠BEC = ∠BCD - ∠CED = ∠ACB - ∠CED = 30° - ∠CED.

Since ∠CAF = 12°,:

∠BAC = ∠CAD + ∠DAF = 18° + 12° = 30°.

Therefore, ∠BCA = 30°, and BC = AC.

Let x = ∠CED. Since BE = CD and BC = AC,: CE = AD = BC = AC.

In ΔCED,: ∠ECD = 180° - ∠CED - ∠CDE = 180° - x - 60° = 120° - x.

In ΔCAD,: ∠CAD + ∠CDA + ∠ACD = 180°, which implies that ∠CDA = 60° - (1/2)∠CAD = 60° - 9° = 51°.

In ΔADF,: ∠ADF = 180° - ∠BAC - ∠DAF = 180° - 30° - 12° = 138°.

In ΔAFE,: ∠AFE = ∠ACB + ∠BEC + ∠CED + ∠ECD + ∠CDA + ∠ADF = 30° + (180° - 30° - x) + x + (120° - x) + 51° + 138° = 489° - x.

Since the angles of a triangle sum to 180°:

∠AFE + ∠EAF + ∠AEF = 180°

∠AFE + 60° + 78° = 180°

∠AFE = 42°.

Therefore, the answer is (A) 42°.

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As a result, 10 individuals purchased οnly Annapurna Pοst, 22 peοple variable and the tοtal number οf persοns whο participated in the pοll was: n = x + b + 34 = 10 + 22 + 34 = 66

A variable is anything that may be altered in the cοntext οf a mathematical nοtiοn οr experiment. Variables are frequently denοted by a single symbοl. The letters x, y, and z are οften used as generic variables symbοls. Variables are qualities with a wide range οf values that may be investigated. Size, age, mοney, where yοu were bοrn, academic pοsitiοn, and kind οf hοusing are just a few examples. Variables may be classified intο twο majοr grοups using bοth numerical and categοrical apprοaches.

k = 100 (100 peοple bοught Kantipur) (100 peοple bοught Kantipur)

a + x = 56 (56 individuals did nοt buy Kantipur) (56 peοple did nοt buy Kantipur)

b = 22 (22 individuals bοught Kantipur alοne) (22 peοple bοught Kantipur οnly)

x + b = n minus (k + a - b) = n minus (100 + (56 - x) - 22) = n minus 34 (With the knοwledge that a + x + b + k - n = 0), such as the elοquence οf the elοquence οf the elοquence οf the elοquence οf the e

n = x + b + 34

a + x = 56 x - an x + 22 = n - k - (a - 22)

x + 22 = x + 22 + a - 100 - a + 22 + 34

x = 10

As a result, 10 individuals purchased οnly Annapurna Pοst, 22 peοple purchased bοth Kantipur and Annapurna Pοst, and the tοtal number οf persοns whο participated in the pοll was:

n = x + b + 34 = 10 + 22 + 34 = 66

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Therefore, the volume V of the solid is 2/3 cubic units.

To find the volume V of the solid, we need to integrate the cross-sectional areas of the isosceles triangles along the x-axis.

Given:

Base of S: Region enclosed by the parabola y = 2 - 2x and the x-axis

Let's denote the variable x as the position along the x-axis.

The height of each isosceles triangle is equal to the base, which is the corresponding value of y on the parabola y = 2 - 2x.

The base of each triangle is the width, which is infinitesimally small dx.

Therefore, the cross-sectional area A at each x position is:

A = (1/2) * base * height

= (1/2) * dx * (2 - 2x)

= dx - dx^2

To find the total volume, we integrate the cross-sectional areas over the region of the base:

V = ∫(A) dx

= ∫(dx - dx^2) from x = 0 to x = 1

Integrating, we get:

V = [x - (1/3)x^3] from x = 0 to x = 1

= (1 - 1/3) - (0 - 0)

= 2/3

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a.  The probability that this runner will complete this road race: In less than 160 minutes is 0.0764. The correct answer is C.

b.  The probability that this runner will complete this road race: In 215 to 245 minutes is 0.1125 The correct answer is C.

a. To find the probability for each scenario, we'll use the given normal distribution parameters:

Mean (μ) = 190 minutes

Standard Deviation (σ) = 21 minutes

Probability of completing the road race in less than 160 minutes:

To calculate this probability, we need to find the area under the normal distribution curve to the left of 160 minutes.

Using the z-score formula: z = (x - μ) / σ

z = (160 - 190) / 21

z ≈ -1.4286

We can then use a standard normal distribution table or statistical software to find the corresponding cumulative probability.

From the standard normal distribution table, the cumulative probability for z ≈ -1.4286 is approximately 0.0764.

Therefore, the probability of completing the road race in less than 160 minutes is approximately 0.0764. The correct answer is C.

b. Probability of completing the road race in 215 to 245 minutes:

To calculate this probability, we need to find the area under the normal distribution curve between 215 and 245 minutes.

First, we calculate the z-scores for each endpoint:

For 215 minutes:

z1 = (215 - 190) / 21

z1 ≈ 1.1905

For 245 minutes:

z2 = (245 - 190) / 21

z2 ≈ 2.6190

Next, we find the cumulative probabilities for each z-score.

From the standard normal distribution table:

The cumulative probability for z ≈ 1.1905 is approximately 0.8820.

The cumulative probability for z ≈ 2.6190 is approximately 0.9955.

To find the probability between these two z-scores, we subtract the cumulative probability at the lower z-score from the cumulative probability at the higher z-score:

Probability = 0.9955 - 0.8820

Probability ≈ 0.1125

Therefore, the probability of completing the road race in 215 to 245 minutes is approximately 0.1125. The correct answer is C.

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Answer:

12 units

Step-by-step explanation:

hope this helped luv!

The series converges by the ratio test, and its sum is given by S = cos1.8 / (1 - cos1.8) = cos1.8 / sin²(0.9) ≈ 19.52.

We can use the ratio test to determine if the series ∑(k=1 to infinity) (cos1.8)^k converges:

Let a_k = (cos1.8)^k.

Then, the ratio of successive terms is:

|a_{k+1}/a_k| = |cos1.8|

Since 0 <= |cos1.8| < 1, the series converges by the ratio test

To find its value, we can use the formula for the sum of an infinite geometric series:

S = a/(1-r)

where a is the first term and

r is the common ratio.

In this case, a = cos1.8 and r = cos1.8.

Thus, the sum of the series is:

S = cos1.8 / (1 - cos1.8) = cos1.8 / sin²(0.9) ≈ 19.52

Therefore, the series converges to approximately 19.52.

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The function rule that represents the situation is: p(t) = 0.9t + 6.75 where t is the number of toppings and p(t) is the total price of the pizza with t toppings.

Why it is?

The function rule represents the situation because it gives the price of a pizza in terms of the number of toppings on the pizza. The fixed cost of the pizza, which is $6.75, is added to the variable cost of the toppings, which is $0.90 per topping. The function takes the number of toppings as an input and outputs the total cost of the pizza.

Therefore, the function rule is a mathematical representation of the relationship between the number of toppings and the price of the pizza.

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The best approach to provide an answer to the question would be: Calculate a confidence interval using t*

The confidence interval is often used to obtain the true population mean. The question asks us to identify evidence that points to the fact that the mean amount of water drank by each teenager is more than 1.5 liters.

We are here searching for the true population mean. We calculate a confidence interval using t when the population variance is not known as is the case here. So, to solve this question, we would calculatethe confidentce interval using t*.

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According to the interpretation of the solution, Dakota's vehicle requires more than 3.6 gallons of fuel to travel the 117 miles to his brother's home.

Dakota will use x gallons of gas to travel 117 miles to his brother's home.

The following inequalities can be used to determine how many gallons of fuel Dakota requires the most:

x ≥ 117/32.5

According to this inequality, x must be more than or equal to 117 miles, which is the required driving distance for Dakota, divided by 32.5 miles per gallon, the maximum fuel efficiency for his vehicle.

After finding x, we obtain:

x ≥ 3.6

Hence, in order to go 117 miles to his brother's home, Dakota would require at least 3.6 gallons of gasoline.

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Conduct additional surveys to increase the total number of responses.

Remove duplicates from the dataset to eliminate skewing of results

Use caution when interpreting the results and consider them exploratory rather than conclusive

Consider using a different methodology for gathering data, such as focus groups or interviews

Adjust the confidence level for the survey results to account for the smaller sample size

Use a different data analysis technique, such as bootstrapping, to account for the limitations of the dataset

Consider using a sample survey method, like stratified sampling or cluster sampling.

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A tile is drawn and replaced, and then a second tile is drawn. Therefore, option A and B are correct answers.

The first two events would be considered independent because the drawing and replacing/removing of one tile does not affect the outcome of the next tile. The third event would not be considered independent because how the first tile is drawn will affect the second one being drawn (since only one of each color is available). The fourth event would also not be considered independent because the outcome of the first tile drawn will affect the second one.

Therefore, option A and B are correct answers.

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To find c, cdf and P(. 1 ≤ X <. 5), the calculation is given.

Suppose that X has the density function f (x) = cx² for 0 ≤ x ≤ 1 and f (x) = 0 otherwise.
a. Find c.
To find c, we need to use the fact that the integral of the density function over the entire range of x should equal 1. That is,
∫0¹ f(x) dx = 1
Substituting the given density function, we get
∫0¹cx² dx = 1
Integrating and simplifying, we get
c(1/3) = 1
Solving for c, we get
c = 3
So, the value of c is 3.
b. Find the cdf.
The cdf is the integral of the density function from the lower limit of x to a given value of x. That is,
F(x) = ∫₀ˣ f(t) dt
Substituting the given density function and simplifying, we get
F(x) = ∫₀ˣ 3t²dt
Integrating and simplifying, we get
F(x) = x³
So the cdf is F(x) = x³
c. What is P(.1 ≤ X < .5)?
To find this probability, we need to find the difference between the cdf at the upper limit and the cdf at the lower limit. That is,
P(.1 ≤ X < .5) = F(.5) - F(.1)
Substituting the cdf that we found earlier, we get
P(.1 ≤ X < .5) = (0.5)³ - (0.1)³
Simplifying, we get
P(.1 ≤ X < .5) = .125 - .001
P(.1 ≤ X < .5) = 0.124
So the probability that X is between .1 and .5 is .124.

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Answer: 21/6

3

18 2/3

Step-by-step explanation:

Answer:

\( \frac{2}{3} (9 - 6) - \frac{5}{6} = \frac{7}{6} \)

\(2(10 - 7) + ( - 3) = 3\)

\(5( \frac{1}{3} \times 4) + 12 = 18 \frac{2}{3} \)

So the correct answers are A for the first problem, B for the second problem, and C for the third problem.

Answer:

y= -3 + 2x

Step-by-step explanation:

If Jenny worked the same shifts as Susan and had the same income of $7,000 last year, then she do not need to file a tax return because  her total income is below the filing threshold.

Whether Jenny needs to file a tax return or not depends on her total income and the types of income she received.

If Jenny had the same income of $7,000 last year as Susan and is single with no dependents,

She will not be required to file a federal tax return for the 2022 tax year as long as her total income is below the filing threshold because for tax year 2022, the filing threshold for single taxpayers under the age of 65 is $12,950.

Therefore , Jenny do not need to file a tax return .

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Answer:

120

Step-by-step explanation:

-5(6) 2(-2)

-5 x6 = -30. 2 x -2=-4

-30 x -4 = 120

the negatives cancel out :)