Find the following product:

To find the product, we will use distributive law as follows:

$\frac{6 x}{5}\left(x^{3}+y^{3}\right)$

$=\frac{6 x}{5} \times x^{3}+\frac{6 x}{5} \times y^{3}$

$=\frac{6}{5} \times\left(x \times x^{3}\right)+\frac{6}{5} \times\left(x \times y^{3}\right)$

$=\frac{6}{5} \times\left(x^{1+3}\right)+\frac{6}{5} \times\left(x \times y^{3}\right)$

$=\frac{6 x^{4}}{5}+\frac{6 x y^{3}}{5}$

Thus, the answer is $\frac{6 x^{4}}{5}+\frac{6 x y^{3}}{5}$.