Question

Find the area of the shaded region.

The area between a rectangle within another rectangle is shaded. The outer rectangle has length 4x and width 2x plus 1 and the inner rectangle has length x plus 7 and width x.
A.

B.

C.

D.

Answer 1

To find the area of the shaded region between two rectangles, we need to first calculate the area of each rectangle and then find the difference between the two areas.

1. **Calculate the area of the outer rectangle:**
The outer rectangle has a length of \( 4x \) and a width of \( 2x + 1 \).

\[
\text{Area of the outer rectangle} = \text{Length} \times \text{Width} = 4x \times (2x + 1)
\]
\[
= 4x(2x) + 4x(1) = 8x^2 + 4x
\]

2. **Calculate the area of the inner rectangle:**
The inner rectangle has a length of \( x + 7 \) and a width of \( x \).

\[
\text{Area of the inner rectangle} = (x + 7) \times x = x^2 + 7x
\]

3. **Find the area of the shaded region:**
The area of the shaded region is the area of the outer rectangle minus the area of the inner rectangle.

\[
\text{Area of the shaded region} = \text{Area of the outer rectangle} - \text{Area of the inner rectangle}
\]
\[
= (8x^2 + 4x) - (x^2 + 7x)
\]
\[
= 8x^2 + 4x - x^2 - 7x
\]
\[
= (8x^2 - x^2) + (4x - 7x)
\]
\[
= 7x^2 - 3x
\]

Thus, the area of the shaded region is:

\[
\boxed{7x^2 - 3x}
\]