Do i use integration by substitution???
Yes you can use a substitution. Rewrite it as $\displaystyle \displaystyle \begin{align*} \frac{1}{3} \int_{-2}^{4}{3x^2 \left( x^3 + 8 \right)^2\, dx} \end{align*}$ then let $\displaystyle \displaystyle \begin{align*} u = x^3 + 8 \implies du = 3x^2 \, dx \end{align*}$ and note that $\displaystyle \displaystyle \begin{align*} u(-2) = 0 \end{align*}$ and $\displaystyle \displaystyle \begin{align*} u(4) = 72 \end{align*}$ and the integral becomes $\displaystyle \displaystyle \begin{align*} \frac{1}{3} \int_0^{72}{u^2\,du} \end{align*}$.
I'm sure you can go from here.