Find the following product:

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

$\frac{4}{3} a\left(a^{2}+b^{2}-3 c^{2}\right)$

$=\frac{4}{3} a \times a^{2}+\frac{4}{3} a \times b^{2}-\frac{4}{3} a \times 3 c^{2}$

$=\frac{4}{3} a^{1+2}+\frac{4}{3} a b^{2}-4 a c^{2}$

$=\frac{4}{3} a^{3}+\frac{4}{3} a b^{2}-4 a c^{2}$

Thus, the answer is $\frac{4}{3} a^{3}+\frac{4}{3} a b^{2}-4 a c^{2}$.