Answer
$$\frac{2\left(2x-y\right)}{x}$$
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:
$$ \frac{\left(x^3+8y^3\right)}{2x^2+5xy+2y^2}\times \frac{8x^2-2y^2}{\left(x^3-2x^2y+4xy^2\right)}\\ \frac{2\left(2x+y\right)\left(2x-y\right)}{x\left(2x+y\right)}\\ \frac{2\left(2x-y\right)}{x}$$