Answer
$$\frac{a-5}{3\left(a-1\right)}$$
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:
$$ \frac{\left(a+2\right)}{a-1}\times \frac{a-5}{\left(3a+6\right)}\\ \frac{\left(a+2\right)\left(a-5\right)}{\left(a-1\right)\left(3a+6\right)}\\ \frac{\left(a+2\right)\left(a-5\right)}{\left(a-1\right)\times \:3\left(a+2\right)}\\ \frac{a-5}{3\left(a-1\right)}$$