Multiply:

To multiply, we will use distributive law as follows:

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

$=3 x^{2}\left(2 x^{2}+3 y^{2}\right)+y^{2}\left(2 x^{2}+3 y^{2}\right)$

$=6 x^{4}+9 x^{2} y^{2}+2 x^{2} y^{2}+3 y^{4}$

$=6 x^{4}+11 x^{2} y^{2}+3 y^{4}$

Thus, the answer is $6 x^{4}+11 x^{2} y^{2}+3 y^{4}$.