Answer
$\frac{4}{x+2}-\frac{4}{(x-1)}+\frac{2}{(x-1)^2}+\frac{1}{(x-1)^3}$
Work Step by Step
Step 1. Assume the end results as: $\frac{A}{x+2}+\frac{B}{(x-1)}+\frac{C}{(x-1)^2}+\frac{D}{(x-1)^3}$
Step 2. Combine the above functions as:
$\frac{A(x-1)^3+B(x+2)(x-1)^2+C(x+2)(x-1)+D(x+2)}{(x+2)(x-1)^3}=\frac{(A+B)x^3+(-3A+C)x^2+(3A-3B+C+D)x+(-A+2B-2C+2D)}{(2x+3)^2}$
Step 3. Compare the above result with the original expression to set up the following system of equations:
\begin{cases} A+B=0\\ -3A+C=-10\\ 3A-3B+C+D=27\\ -A+2B-2C+2D=-14 \end{cases}
Step 4. Use substitutions $B=-A, C=3A-10$ to get
\begin{cases} 3A+3A+(3A-10)+D=27\\ -A-2A-2(3A-10)+2D=-14 \end{cases} or \begin{cases} 9A+D=37\\ 9A-2D=34 \end{cases}
Step 5. Solve the equations to get $A=4, B=-4, C=2, D=1$
Step 6. Write the final results as:
$\frac{4}{x+2}-\frac{4}{(x-1)}+\frac{2}{(x-1)^2}+\frac{1}{(x-1)^3}$
