Answer
$-\frac{2}{x+2}+\frac{2}{x}-\frac{1}{x^3}$
Work Step by Step
Step 1. Factor the denominator as: $(x+2)x^3$
Step 2. Assume the end results as: $\frac{A}{x+2}+\frac{B}{x}+\frac{C}{x^2}+\frac{D}{x^3}$
Step 3. Combine the above functions as:
$\frac{Ax^3+B(x+2)x^2+C(x+2)x+D(x+2)}{(x+2)x^3}=\frac{(A+B)x^3+(2B+C)x^2+(2C+D)x+(2D)}{(x+2)x^3}$
Step 4. Compare the above result with the original expression to set up the following system of equations:
\begin{cases} A+B=0 \\ 2B+C=4\\ 2C+D=-1\\ 2D=-2 \end{cases}
Step 5. Use back substitution to solve the above equations and get $A=-2, B=2,C=0,D=-1$
Step 6. Write the final results as:
$-\frac{2}{x+2}+\frac{2}{x}-\frac{1}{x^3}$
