Answer
$\frac{4}{x-1}-\frac{2}{(x-1)^2}-\frac{4}{x}$
Work Step by Step
Step 1. Assume the partial fraction decomposition is:
$\frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$
Step 2. Combine the rational functions above:
$\frac{A(x-1)^2+Bx(x-1)+Cx}{x(x-1)^2}\frac{(A+B)x^2+(-2A-B+C)x+(A)}{x(x-1)^2}$
Step 4. Compare the above function with the original to setup the following system of equations:
$\begin{cases} A+B=0\\-2A-B+C=2\\A=-4 \end{cases}$
Step 5. Solve the above equations: $A=-4, B=-A=4, C=2+2A+B=-2$
Step 6. The final result is:
$\frac{-4}{x}+\frac{4}{x-1}+\frac{-2}{(x-1)^2}$ or $\frac{4}{x-1}-\frac{2}{(x-1)^2}-\frac{4}{x}$
