Answer
$$-\frac{3}{\left(x+2\right)^2} +\frac{5}{x+2} +\ln \left|x+2\right|+C$$
Work Step by Step
Given
$$ \int \frac{\left(x^{2}-x\right) d x}{(x+2)^{3}} $$
Since
\begin{align*} \frac{\left(x^{2}-x\right) }{(x+2)^{3}}&= \frac{A}{(x+2)^3}+ \frac{B}{(x+2)^2}+ \frac{C}{(x+2) }\\ &=\frac{ A+B(x+2)+C(x+2)^2 }{(x+2)^{3}}\\ x^2-x&= A+B(x+2)+C(x+2)^2 \end{align*}
Then by comparing the coefficients, we get
$$ C= 1,\ \ B=-5,\ \ A=6$$
Hence
\begin{align*}
\int \frac{\left(x^{2}-x\right) }{(x+2)^{3}}dx&=\int \frac{6}{(x+2)^3}dx+ \int \frac{-5}{(x+2)^2}dx+ \int\frac{1}{(x+2) }dx \\ &=-\frac{3}{\left(x+2\right)^2} +\frac{5}{x+2} +\ln \left|x+2\right|+C
\end{align*}
