Answer
$$\frac{\left(n+3\right)^2}{n-2}$$
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:
$$ \frac{\left(n^2+5n+6\right)\left(n+3\right)}{n^2-4}\\ \frac{\left(n+2\right)\left(n+3\right)^2}{\left(n+2\right)\left(n-2\right)}\\ \frac{\left(n+3\right)^2}{n-2}$$
