Find each of the following product:

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

We have:

$\left(-4 x^{2}\right) \times\left(-6 x y^{2}\right) \times\left(-3 y z^{2}\right)$

$=\{(-4) \times(-6) \times(-3)\} \times\left(x^{2} \times x\right) \times\left(y^{2} \times y\right) \times z^{2}$

$=\{(-4) \times(-6) \times(-3)\} \times\left(x^{2+1}\right) \times\left(y^{2+1}\right) \times z^{2}$

$=-72 x^{3} y^{3} z^{2}$

Thus, the answer is $-72 x^{3} y^{3} z^{2}$.