Find the following product:

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

$\frac{7}{5} x^{2} y\left(\frac{3}{5} x y^{2}+\frac{2}{5} x\right)$

$=\frac{7}{5} x^{2} y \times \frac{3}{5} x y^{2}+\frac{7}{5} x^{2} y \times \frac{2}{5} x$

$=\frac{21}{25} x^{2+1} y^{1+2}+\frac{14}{25} x^{2+1} y$

$=\frac{21}{25} x^{3} y^{3}+\frac{14}{25} x^{3} y$

Thus, the answer is $\frac{21}{25} x^{3} y^{3}+\frac{14}{25} x^{3} y$.