$$\frac{a^{-4}-9b^{-2}}{a^{-2}-3b^{-1}}·\frac{{\left(b+3a^2\right)}^{-1}}{a^{-2}{b}^{-1}}$$

$$=\frac{\left(a^{-2}-3b^{-1}\right)·\left(a^{-2}+3b^{-1}\right)}{{\displaystyle \frac{1}{a^2}}-3{\displaystyle \frac{1}{b}}}·\frac{{\displaystyle \frac{1}{b+3a^2}}}{{\displaystyle \frac{1}{a^2}}·{\displaystyle \frac{1}{b}}}$$

$$=\frac{\cancel{\left({\displaystyle \frac{1}{a^2}}-3{\displaystyle \frac{1}{b}}\right)}·\left({\displaystyle \frac{1}{a^2}}+3{\displaystyle \frac{1}{b}}\right)}{\cancel{{\displaystyle \frac{1}{a^2}}-3{\displaystyle \frac{1}{b}}}}·\frac{1}{b+3a^2}÷\frac{1}{a^{2}b}$$

$$=\left(\frac{1}{a^2}+\frac{3}{b}\right)·\frac{1}{b+3a^2}·\frac{a^{2}b}{1}$$

$$=\left(\frac{1}{a^2}·\frac{b}{b}+\frac{3}{b}·\frac{a^2}{a^2}\right)·\frac{a^{2}b}{b+3a^2}$$

$$=\left(\frac{1b}{a^{2}b}+\frac{3a^2}{a^{2}b}\right)·\frac{a^{2}b}{b+3a^2}$$

$$=\frac{\cancel{b+3a^2}}{\bcancel{a^{2}b}}·\frac{\bcancel{a^{2}b}}{\cancel{b+3a^2}}$$