Find the following product:

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

$-11 y^{2}(3 y+7)$

$=\left(-11 y^{2}\right) \times 3 y+\left(-11 y^{2}\right) \times 7$

$=(-11 \times 3)\left(y^{2} \times y\right)+(-11 \times 7) \times\left(y^{2}\right)$

$=(-33)\left(y^{2+1}\right)+(-77) \times\left(y^{2}\right)$

$=-33 y^{3}-77 y^{2}$

Thus, the answer is $-33 y^{3}-77 y^{2}$.