Answer
$$\frac{t+1}{4\left(t+2\right)}$$
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:
$$ \frac{\left(t-3\right)}{t+2}\times \frac{t+1}{\left(4t-12\right)}\\ \frac{\left(t-3\right)\left(t+1\right)}{\left(t+2\right)\left(4t-12\right)}\\ \frac{\left(t-3\right)\left(t+1\right)}{\left(t+2\right)\times \:4\left(t-3\right)}\\ \frac{t+1}{4\left(t+2\right)}$$