Answer
$$\frac{x^2+4x+16}{\left(x+4\right)^2}$$
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:
$$ \frac{\left(x^3-64\right)}{x^3+64}\times \frac{x^2-4x+16}{\left(x^2-16\right)}\\ \frac{\left(x-4\right)\left(x^2+4x+16\right)\left(x^2-4x+16\right)}{\left(x^2-4x+16\right)\left(x+4\right)^2\left(x-4\right)} \\ \frac{x^2+4x+16}{\left(x+4\right)^2}$$