Answer
$$\frac{4\left(t-3\right)}{3\left(t+1\right)}$$
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:
$$ \frac{t-3}{6}\div \frac{\left(t+1\right)}{8}\\ \frac{t-3}{6}\times \frac{8}{\left(t+1\right)}\\ \frac{\left(t-3\right)\times \:8}{6\left(t+1\right)}\\ \frac{4\left(t-3\right)}{3\left(t+1\right)}$$