#### Answer

$-\frac{3a^{3}}{b}$

#### Work Step by Step

Since all terms are multiplied by each other, we can collect constants and variables separately in their own parentheses. We then use the rules $a^{m}\times a^{n}=a^{m+n}$ and $\frac{1}{x^2}= x^{-2}$ to simplify the expression:
$\frac{-12a^{-2}b^{3}}{4a^{-5}b^{4}}$
=$(\frac{-12}{4})\times(\frac{a^{-2}}{a^{-5}})\times(\frac{b^{3}}{b^{4}})$
=$(-3)\times(a^{-2-(-5)})\times(b^{3-4})$
=$(-3)\times(a^{-2+5})\times(b^{-1})$
=$(-3)\times(a^{3})\times(b^{-1})$
=$(-3)\times(a^{3})\times(\frac{1}{b})$
=$-\frac{3a^{3}}{b}$