Question

Use the image to answer the question.

A rectangle has its length labeled x-squared minus x plus 1 and its width labeled 2 x-squared plus 3 x minus 2.

A poster board for a science project is in the shape of a rectangle. Determine the polynomial expression that represents the perimeter of the poster board.

(1 point)

Answer 1

To find the polynomial expression that represents the perimeter of the rectangular poster board, we use the formula for the perimeter \( P \) of a rectangle, which is given by:

\[
P = 2(\text{length} + \text{width})
\]

In this case, the length \( L \) is given as:

\[
L = x^2 - x + 1
\]

And the width \( W \) is given as:

\[
W = 2x^2 + 3x - 2
\]

First, we will add the length and width together:

\[
L + W = (x^2 - x + 1) + (2x^2 + 3x - 2)
\]

Now combine like terms:

\[
L + W = x^2 + 2x^2 - x + 3x + 1 - 2
\]
\[
= (1 + 2)x^2 + (-1 + 3)x + (1 - 2)
\]
\[
= 3x^2 + 2x - 1
\]

Next, we substitute this expression back into the perimeter formula:

\[
P = 2(L + W) = 2(3x^2 + 2x - 1)
\]

Now, distribute the 2:

\[
P = 2 \cdot 3x^2 + 2 \cdot 2x + 2 \cdot (-1)
\]
\[
= 6x^2 + 4x - 2
\]

Thus, the polynomial expression that represents the perimeter of the poster board is:

\[
\boxed{6x^2 + 4x - 2}
\]