Question

margot is sewing a ribbon on a seam along the perimeter of a square pillow. the side length of the pillow is $2x^2 + 1$ inches. she plans to make a similar pillow, including the ribbon, whose side length is $4x - 7$ inches. what expression can be used for the length of ribbon that she needs for both pillows, and what is the length if $x = 3.5$?
$2x^2 + 4x - 6$; 32.5 inches
$4(2x^2 + 4x - 6)$; 130.0 inches
$2x^2 + 4x - 6$; 22.0 inches
$4(2x^2 + 4x - 6)$; 88.0 inches

Explanation:

Step1: Find perimeter of first pillow

Perimeter of square = $4\times$ side length.
First pillow side: $2x^2 + 1$, so perimeter: $4(2x^2 + 1) = 8x^2 + 4$

Step2: Find perimeter of second pillow

Second pillow side: $4x - 7$, so perimeter: $4(4x - 7) = 16x - 28$

Step3: Sum perimeters for total ribbon

Add the two perimeters:
$$(8x^2 + 4) + (16x - 28) = 8x^2 + 16x - 24 = 4(2x^2 + 4x - 6)$$

Step4: Substitute $x=3.5$ into expression

First calculate inside the parentheses:
$2(3.5)^2 + 4(3.5) - 6 = 2(12.25) + 14 - 6 = 24.5 + 14 - 6 = 32.5$
Multiply by 4: $4\times32.5 = 130.0$

Answer:

4(2x² + 4x − 6); 130.0 inches