Answer
$$\frac{5b^2+4ba}{2\left(a+b\right)}$$
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:
$$ \frac{\left(8a^3+b^3\right)}{2a^2+3ab+b^2}\times \frac{4b^2+4ab+b^2}{\left(8a^2-4ab+2b^2\right)}\\ \frac{\left(4a^2-2ab+b^2\right)b\left(5b+4a\right)}{\left(a+b\right)\times \:2\left(4a^2-2ab+b^2\right)}\\ \frac{5b^2+4ba}{2\left(a+b\right)}$$