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:

$\left(-\frac{2}{7} a^{4}\right) \times\left(-\frac{3}{4} a^{2} b\right) \times\left(-\frac{14}{5} b^{2}\right)$

$=\left\{\left(-\frac{2}{7}\right) \times\left(-\frac{3}{4}\right) \times\left(-\frac{14}{5}\right)\right\} \times\left(a^{4} \times a^{2}\right) \times\left(b \times b^{2}\right)$

$=\left\{-\left(\frac{2}{7} \times \frac{3}{4} \times \frac{14}{5}\right)\right\} \times a^{4+2} \times b^{1+2}$

$=-\frac{3}{5} a^{6} b^{3}$

Thus, the answer is $-\frac{3}{5} a^{6} b^{3}$.