Answer
$$\frac{4x^2+18x}{\left(x+4\right)\left(x+5\right)}$$
Work Step by Step
The perimeter of a rectangle is equal to the sum of twice the length of each of the adjacent sides. Doing this, we find:
$$\frac{2x}{x+4}+\frac{2x}{x+5}\\ \frac{2x\left(x+5\right)+2x\left(x+4\right)}{\left(x+4\right)\left(x+5\right)} \\ \frac{4x^2+18x}{\left(x+4\right)\left(x+5\right)}$$
