Answer
The required solution is
$\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)}$
Work Step by Step
We know that the perimeter of the rectangle can be calculated by using the formula:
$P=2l+2b$
Here, $l$ refers to the length and $b$ refers to the width of the rectangle.
Put $l=\frac{x}{x+5}$ and $b=\frac{x}{x+6}$ in the equation of $P$.
$\begin{align}
& P=2\left( \frac{x}{x+5} \right)+2\left( \frac{x}{x+6} \right) \\
& =\frac{2x}{x+5}+\frac{2x}{x+6}
\end{align}$
And solve the above rational expression by taking the least common denominator.
$\begin{align}
& P=\frac{2x}{x+5}\times \frac{\left( x+6 \right)}{\left( x+6 \right)}+\frac{2x}{x+6}\times \frac{\left( x+5 \right)}{\left( x+5 \right)} \\
& =\frac{2x\left( x+6 \right)}{\left( x+5 \right)\left( x+6 \right)}+\frac{2x\left( x+5 \right)}{\left( x+6 \right)\left( x+5 \right)} \\
& =\frac{2{{x}^{2}}+12x+2{{x}^{2}}+10x}{\left( x+5 \right)\left( x+6 \right)} \\
& =\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)}
\end{align}$
Hence, the perimeter of the rectangle as a single rational expression is $\frac{4{{x}^{2}}+22x}{\left( x+5 \right)\left( x+6 \right)}$.
