Multiply:

To multiply, we will use distributive law as follows:

$\left(3 x^{2} y-5 x y^{2}\right)$ by $\left(\frac{1}{5} x^{2}+\frac{1}{3} y^{2}\right)$

$=\frac{1}{5} x^{2}\left(3 x^{2} y-5 x y^{2}\right)+\frac{1}{3} y^{2}\left(3 x^{2} y-5 x y^{2}\right)$

$=\frac{3}{5} x^{4} y-x^{3} y^{2}+x^{2} y^{3}-\frac{5}{3} x y^{4}$

Thus, the answer is $\frac{3}{5} x^{4} y-x^{3} y^{2}+x^{2} y^{3}-\frac{5}{3} x y^{4}$.