Find the following product:

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

$2 a^{3}(3 a+5 b)$

$=2 a^{3} \times 3 a+2 a^{3} \times 5 b$

$=(2 \times 3)\left(a^{3} \times a\right)+(2 \times 5) a^{3} b$

$=(2 \times 3) a^{3+1}+(2 \times 5) a^{3} b$

$=6 a^{4}+10 a^{3} b$

Thus, the answer is $6 a^{4}+10 a^{3} b$.