Question

1. write an expression to represent the area of the shaded region in simplest form.

diagram: larger rectangle (shaded) with height ( 5x + 2 ), width ( 3x - 1 ); smaller unshaded rectangle with height ( x ), width ( x + 7 ). handwritten notes and other text present but not part of the core problem.
© gina wilson (all things algebra®, llc), 2012-2017

Explanation:

Step1: Find area of outer rectangle

The outer rectangle has length \( 3x - 1 \) and width \( 5x + 2 \). The area of a rectangle is length times width, so:
\( (3x - 1)(5x + 2) \)
Using the distributive property (FOIL method):
\( 3x \times 5x + 3x \times 2 - 1 \times 5x - 1 \times 2 = 15x^2 + 6x - 5x - 2 = 15x^2 + x - 2 \)

Step2: Find area of inner rectangle

The inner rectangle has length \( x + 7 \) and width \( x \). The area is:
\( x(x + 7) = x^2 + 7x \)

Step3: Subtract inner area from outer area

The area of the shaded region is outer area minus inner area:
\( (15x^2 + x - 2) - (x^2 + 7x) \)
Distribute the negative sign:
\( 15x^2 + x - 2 - x^2 - 7x \)
Combine like terms:
\( (15x^2 - x^2) + (x - 7x) - 2 = 14x^2 - 6x - 2 \)

Answer:

\( 14x^2 - 6x - 2 \)