Question

find the area of the shaded region.
8x + 2
4x + 7
3x
x + 5

Explanation:

Step1: Find area of outer rectangle

The formula for the area of a rectangle is \( \text{length} \times \text{width} \). The outer rectangle has length \( 8x + 2 \) and width \( 4x + 7 \). So its area is \( (8x + 2)(4x + 7) \).
Expanding this: \( 8x \times 4x + 8x \times 7 + 2 \times 4x + 2 \times 7 = 32x^2 + 56x + 8x + 14 = 32x^2 + 64x + 14 \).

Step2: Find area of inner rectangle

The inner rectangle has length \( 3x \) and width \( x + 5 \). Its area is \( 3x(x + 5) \).
Expanding this: \( 3x \times x + 3x \times 5 = 3x^2 + 15x \).

Step3: Subtract inner area from outer area

The shaded area is the area of the outer rectangle minus the area of the inner rectangle. So we calculate \( (32x^2 + 64x + 14) - (3x^2 + 15x) \).
Simplifying: \( 32x^2 - 3x^2 + 64x - 15x + 14 = 29x^2 + 49x + 14 \).

Answer:

The area of the shaded region is \( 29x^2 + 49x + 14 \).