Find the following product:

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

1. $5 x\left(10 x^{2} y-100 x y^{2}\right)$\

$=\left(1.5 x \times 10 x^{2} y\right)-\left(1.5 x \times 100 x y^{2}\right)$

$=\left(15 x^{1+2} y\right)-\left(150 x^{1+1} y^{2}\right)$

$=15 x^{3} y-150 x^{2} y^{2}$

Thus, the answer is $15 x^{3} y-150 x^{2} y^{2}$.