Question

Find the area of the rectangle.

Short answer

Option ($\mathit{J}$) is the correct answer.

Step-by-step solution

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$ and $q$ are the polynomials and $q\ne 0$.

Step 2. Define the area of the rectangle.

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

Step 3. Define the product of two expressions ab and cd.

The product of two expressions $\frac{a}{b}$and $\frac{c}{d}$, where $b\ne 0$and $d\ne 0$, is given below.

$\frac{a}{b}.\frac{c}{d}=\frac{ac}{bd}\mathrm{...........}\left(2\right)$

Step 4. Calculate the area of the given rectangle.

Observe the figure.

The length of the rectangle $=\frac{x^2-4}{x+3}$

The width of the rectangle role="math" localid="1648028748102" $=\frac{x+3}{x+2}$

Use $\left(1\right)$to find the area of the rectangle.

The area of the rectangle is

$A=\left(\frac{{\displaystyle x^2-4}}{{\displaystyle x+3}}\right)\left(\frac{{\displaystyle x^2-4}}{{\displaystyle x+3}}\right)$

Use $\left(2\right)$to find the product.

role="math" localid="1648028852316" $\begin{array}{l}A=\left(\frac{x^2-4}{x+3}\right)\left(\frac{x+3}{x+2}\right)\\ A=\frac{(x^2-4)(x+3)}{(x+3)(x+2)}\\ =\frac{x^2-4}{x+2}\\ =\frac{x^2-2^2}{x+2}\\ =\frac{(x+2)(x-2)}{(x+2)}\\ =x-2\end{array}$

So, the area of the given rectangle is $x-2$.

Therefore option $\left(J\right)$is the correct answer.