Question

Use the image to answer the question.

A rectangle has its length labeled x-squared plus 2 x plus 4 and its width labeled 5 x minus 3.

The dimensions of a rectangular stage are shown. Determine the polynomial expression that represents the area of the stage.

(1 point)

Answer 1

To find the area of a rectangle, you can use the formula:

\[
\text{Area} = \text{Length} \times \text{Width}
\]

In this case, the length is given by the polynomial \(x^2 + 2x + 4\) and the width is given by \(5x - 3\).

To find the area, we will multiply these two polynomials:

\[
\text{Area} = (x^2 + 2x + 4)(5x - 3)
\]

Now we will use the distributive property (also known as the FOIL method for binomials) to multiply the two expressions:

1. Multiply \(x^2\) by \(5x\) and \(-3\):
\[
x^2 \cdot 5x = 5x^3
\]
\[
x^2 \cdot (-3) = -3x^2
\]

2. Multiply \(2x\) by \(5x\) and \(-3\):
\[
2x \cdot 5x = 10x^2
\]
\[
2x \cdot (-3) = -6x
\]

3. Multiply \(4\) by \(5x\) and \(-3\):
\[
4 \cdot 5x = 20x
\]
\[
4 \cdot (-3) = -12
\]

Now we can combine all these results:

\[
\text{Area} = 5x^3 + (-3x^2 + 10x^2) + (-6x + 20x) - 12
\]
\[
= 5x^3 + 7x^2 + 14x - 12
\]

Thus, the polynomial expression that represents the area of the stage is:

\[
\boxed{5x^3 + 7x^2 + 14x - 12}
\]