Answer
$$\frac{1}{2} \ln |2 x+5|+\frac{5}{(2 x+5)}-\frac{5}{4(2 x+5)^{2}}+C$$
Work Step by Step
Given $$\int \frac{4 x^{2}-20}{(2 x+5)^{3}} d x$$
Since
\begin{aligned}
\frac{4 x^{2}-20}{(2 x+5)^{3}} &=\frac{\mathrm{A}}{2x+5}+\frac{\mathrm{B}}{(2x+5)^2}+\frac{\mathrm{C}}{(2x+5)^3} \\
&=\frac{A(2 x+5)^{2}+B(2 x+5)+C}{(2 x+5)^{3}} \\
4 x^{2}-20&= A(2 x+5)^{2}+B(2 x+5)+C
\end{aligned}
By comparing coefficients, we get
\begin{align*}
A&=1\\
B&=-10\\
C&=5
\end{align*}
Hence
\begin{aligned}
\int \frac{4 x^{2}-20}{(2 x+5)^{3}} d x&=\int \frac{1}{2x+5}dx-10\int \frac{1}{(2x+5)^2}+5\int \frac{1}{(2x+5)^3}dx\\
&= \frac{1}{2} \ln |2 x+5|+\frac{5}{(2 x+5)}-\frac{5}{4(2 x+5)^{2}}+C
\end{aligned}
