Question

Write an expression to represent the area of the shaded region.

Find the area of the rectangle.

Short answer

The area of the shaded region is $\left(3x^2-21\right)$units².

Step-by-step solution

Step 1. Define an expression.

A mathematical expression is a combinations of numbers and variables ordered with mathematical operations.

Step 2. Define the area of the rectangle.

Area of the rectangle $\left(A\right)=$ length$\times$width ……….. (1)

Step 3. Expand the identity, (a+b)(a−b).

$\left(a+b\right)\left(a-b\right)=a^2-b^2\mathrm{...........}\left(2\right)$

Step 4. Calculate the area of the shaded region.

Observe the figure.

The length of the outer rectangle$=2x+5$ units

The width of the outer rectangle$=2x-5$ units

The length of the inner rectangle$=x+2$ units

The width of the inner rectangle$=x-2$ units

Use equation (1).

The area of the outer rectangle is

$A_o=\left(2x+5\right)\left(2x-5\right)$

Use equation (2) to find the product.

$\begin{array}{c}{A}_o={\left(2x\right)}^2-5^2\\ A_o={\left(2\right)}^2{\left(x\right)}^2-25\\ A_o=4x^2-25\end{array}$

Use equation (1).

The area of inner rectangle is

$A_i=\left(x+2\right)\left(x-2\right)$

Use equation (2) to find the product.

$\begin{array}{c}{A}_i=x^2-2^2\\ A_i=x^2-4\end{array}$

Since, the area of the shaded region area of the outer rectangle$-$area of inner rectangle

So, the area of the shaded region is

$\begin{array}{c}=\left(4x^2-25\right)-\left(x^2-4\right)\\ =4x^2-25-x^2+4\\ =4x^2-x^2-25+4\\ =3x^2-21\end{array}$

Therefore, the area of the shaded region is$\left(3x^2-21\right)$ units².