Find each of the following product:

To multiply algebraic expressions, we can use commutative and associative laws along with the law of indices, $a^{m} \times a^{n}=a^{m+n}$, wherever applicable.

We have:

$-3 a^{2} \times 4 b^{4}$

$=(-3 \times 4) \times\left(a^{2} \times b^{4}\right)$

$=-12 a^{2} b^{4}$

Thus, the answer is $-12 a^{2} b^{4}$.