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