91Ó°ÊÓ

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

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

$\left(a+b\right)\left(a−b\right)=a^2−b^2\text{ â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰â¶Ä‰}\mathrm{...........}\left(2\right)$

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.