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. 6x2-3x

B. 6x2+11x

C. 7x2-3x

D. 7x2+11x

Answer 1

To find the area of the shaded region, we need to calculate the area of both the outer rectangle and the inner rectangle, and then subtract the area of the inner rectangle from the area of the outer rectangle.

### Step 1: Calculate the area of the outer rectangle.
The dimensions of the outer rectangle are:
- Length = \(4x\)
- Width = \(2x + 1\)

The area of the outer rectangle (A_outer) is given by:
\[
A_{\text{outer}} = \text{length} \times \text{width} = 4x \times (2x + 1)
\]

Distributing:
\[
A_{\text{outer}} = 4x(2x) + 4x(1) = 8x^2 + 4x
\]

### Step 2: Calculate the area of the inner rectangle.
The dimensions of the inner rectangle are:
- Length = \(x + 7\)
- Width = \(x\)

The area of the inner rectangle (A_inner) is given by:
\[
A_{\text{inner}} = \text{length} \times \text{width} = (x + 7) \times x
\]

Distributing:
\[
A_{\text{inner}} = x^2 + 7x
\]

### Step 3: Calculate the area of the shaded region.
The area of the shaded region (A_shaded) is given by:
\[
A_{\text{shaded}} = A_{\text{outer}} - A_{\text{inner}}
\]

Substituting the areas we calculated:
\[
A_{\text{shaded}} = (8x^2 + 4x) - (x^2 + 7x)
\]

Combining like terms:
\[
A_{\text{shaded}} = 8x^2 + 4x - x^2 - 7x
\]
\[
A_{\text{shaded}} = (8x^2 - x^2) + (4x - 7x)
\]
\[
A_{\text{shaded}} = 7x^2 - 3x
\]

### Conclusion
Thus, the area of the shaded region is:
\[
\boxed{7x^2 - 3x}
\]

The correct option is **C. 7x² - 3x**.