Answer
$ \frac{2x(2x+9)}{(x+4)(x+5)} $.
Work Step by Step
The perimeter of the rectangle is $=2(sum\; of \; the \;sides)$.
The given sides of the rectangle are:
$\Rightarrow \frac{x}{x+5}$ and $\frac{x}{x+4}$
Thus, the perimeter of the rectangle is:
$\Rightarrow 2\left ( \frac{x}{x+5}+\frac{x}{x+4} \right )$
Factor out $x$.
$\Rightarrow 2x\left ( \frac{1}{x+5}+\frac{1}{x+4} \right )$
The LCD of the denominators is $(x+5)(x+4)$.
Multiply both the numerator and the denominator:
$\Rightarrow 2x\left ( \frac{x+4}{(x+4)(x+5)}+\frac{x+5}{(x+4)(x+5)} \right )$
Add both the numerators because denominators are equal.
$\Rightarrow 2x\left ( \frac{x+4+x+5}{(x+4)(x+5)} \right )$
Simplify.
$\Rightarrow 2x\left ( \frac{2x+9}{(x+4)(x+5)} \right )$
$\Rightarrow \frac{2x(2x+9)}{(x+4)(x+5)} $.
