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:

$\left(-\frac{1}{27} a^{2} b^{2}\right) \times\left(\frac{9}{2} a^{3} b^{2} c^{2}\right)$

$=\left(-\frac{1}{27} \times \frac{9}{2}\right) \times\left(a^{2} \times a^{3}\right) \times\left(b^{2} \times b^{2}\right) \times c^{2}$

$=\left(-\frac{1}{27} \times \frac{9}{2}\right) \times\left(a^{2+3}\right) \times\left(b^{2+2}\right) \times c^{2}$

$=-\frac{1}{6} a^{5} b^{4} c^{2}$

Thus, the answer is $-\frac{1}{6} a^{5} b^{4} c^{2}$.