Multiply:

To multiply the expressions, we will use the distributive law in the following way:

$\left(\frac{x}{7}+\frac{x^{2}}{2}\right) b y\left(\frac{2}{5}+\frac{9 x}{4}\right)$

$=\frac{x}{7}\left(\frac{2}{5}+\frac{9 x}{4}\right)+\frac{x^{2}}{2}\left(\frac{2}{5}+\frac{9 x}{4}\right)$

$=\frac{2 x}{35}+\frac{9 x^{2}}{28}+\frac{x^{2}}{5}+\frac{9 x^{3}}{8}$

$=\frac{2 x}{35}+\left(\frac{45+28}{140}\right) x^{2}+\frac{9 x^{3}}{8}$

$=\frac{2 x}{35}+\frac{73 x^{2}}{140}+\frac{9 x^{2}}{8}$

Thus, the answer is $\frac{2 x}{35}+\frac{73 x^{2}}{140}+\frac{9 x^{3}}{8}$