We going to use the AROC formula
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
Being a=-2 and b=-4
Let's find the numerator
[tex]\begin{gathered} f(b)=-3\left(-4\right)^3+12(-4)-21 \\ f(b)=123 \end{gathered}[/tex][tex]\begin{gathered} f(a)=-3\left(-2\right)^3+12(-2)-21 \\ f(a)=-21 \end{gathered}[/tex]
Replacing
[tex]\frac{123-(-21)}{-4-(-2)}=-\frac{144}{2}=-72[/tex]
Answer: -72
