Answer
$$\frac{2x^2+9x+9}{\left(x+1\right)^2}$$
Work Step by Step
Recall, one never divides by a fraction. Rather, if we want to divide by a fraction, we multiply by the reciprocal of the fraction. Recall, the reciprocal of a fraction is that fraction with the numerator and the denominator switched. Thus, we find:
$$ \frac{\left(x-2\right)\left(2x+3\right)}{\left(x+1\right)\left(x-5\right)}\times \frac{\left(x-5\right)\left(x+3\right)}{\left(x-2\right)\left(x+1\right)}\\ \frac{\left(2x+3\right)\left(x+3\right)}{\left(x+1\right)^2}\\ \frac{2x^2+9x+9}{\left(x+1\right)^2}$$
