Find the following product:

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

$-\frac{4}{27} x y z\left(\frac{9}{2} x^{2} y z-\frac{3}{4} x y z^{2}\right)$

$=\left\{\left(-\frac{4}{27} x y z\right)\left(\frac{9}{2} x^{2} y z\right)\right\}-\left\{\left(-\frac{4}{27} x y z\right)\left(\frac{3}{4} x y z^{2}\right)\right\}$

$=\left\{\left(-\frac{4}{27} \times \frac{9}{2}\right)\left(x^{1+2} y^{1+1} z^{1+1}\right)\right\}-\left\{\left(-\frac{4}{27} \times \frac{3}{4}\right)\left(x^{1+1} y^{1+1} z^{1+2}\right)\right\}$

$=-\frac{2}{3} x^{3} y^{2} z^{2}+\frac{1}{9} x^{2} y^{2} z^{3}$

Thus, the answer is $-\frac{2}{3} x^{3} y^{2} z^{2}+\frac{1}{9} x^{2} y^{2} z^{3}$.