Answer
$\dfrac{2}{(x+2)^2}-\dfrac{3}{(x+2)^3}$
Work Step by Step
We will write the partial decomposition as:
$\dfrac{2x+1}{(x+2)^3}=\dfrac{A}{x+2}+\dfrac{B}{(x+2)^2}+\dfrac{C}{(x+2)^3} ~~~~(a)$
This implies that $2x+1=A(x+2)^2+B(x+2)+C$
Plug $x=-3,0$ to obtain: $A -B=-2$ and $2A+B=2$
After solving these equations, we get: $A=0$ and $B=2$
Now, plug $x=-2$ to obtain: $C=-3$
So, the equation (a) becomes:
$\dfrac{2x+1}{(x+2)^3}=\dfrac{2}{(x+2)^2}-\dfrac{3}{(x+2)^3}$
