Answer
The required solution is
$\frac{4{{x}^{2}}+14x}{\left( x+3 \right)\left( x+4 \right)}\text{ }\text{.}$
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 $l=\frac{x}{x+3}$ and $b=\frac{x}{x+4}$ in the equation of $P$.
$\begin{align}
& P=2\left( \frac{x}{x+3} \right)+2\left( \frac{x}{x+4} \right) \\
& =\frac{2x}{x+3}+\frac{2x}{x+4}
\end{align}$.
The required solution is
$\frac{4{{x}^{2}}+14x}{\left( x+3 \right)\left( x+4 \right)}\text{ }\text{.}$
Solve the above rational expression by taking the least common denominator:
$\begin{align}
& P=\frac{2x}{x+3}\times \frac{\left( x+4 \right)}{\left( x+4 \right)}+\frac{2x}{x+4}\times \frac{\left( x+3 \right)}{\left( x+3 \right)} \\
& =\frac{2x\left( x+4 \right)}{\left( x+3 \right)\left( x+4 \right)}+\frac{2x\left( x+3 \right)}{\left( x+4 \right)\left( x+3 \right)} \\
& =\frac{2{{x}^{2}}+8x+2{{x}^{2}}+6x}{\left( x+3 \right)\left( x+4 \right)} \\
& =\frac{4{{x}^{2}}+14x}{\left( x+3 \right)\left( x+4 \right)}
\end{align}$.
Therefore, the perimeter of the rectangle as a single rational expression is $\frac{4{{x}^{2}}+14x}{\left( x+3 \right)\left( x+4 \right)}$.
