Multiply:

To multiply, we will use distributive law as follows:

$\left(-\frac{a}{7}+\frac{a^{2}}{9}\right) b y\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$

$=\left(-\frac{a}{7}\right)\left(\frac{b}{2}-\frac{b^{2}}{3}\right)+\left(\frac{a^{2}}{9}\right)\left(\frac{b}{2}-\frac{b^{2}}{3}\right)$

$=\left(-\frac{a b}{14}+\frac{a b^{2}}{21}\right)+\left(\frac{a^{2} b}{18}-\frac{a^{2} b^{2}}{27}\right)$

$=-\frac{a b}{14}+\frac{a b^{2}}{21}+\frac{a^{2} b}{18}-\frac{a^{2} b^{2}}{27}$

Thus, the answer is $-\frac{a b}{14}+\frac{a b^{2}}{21}+\frac{a^{2} b}{18}-\frac{a^{2} b^{2}}{27}$.