Answer
$-\frac{135}{64(3x+2)}+\frac{109}{64(x-2)}+\frac{9}{8(x-2)^2}$
Work Step by Step
Step 1. Factor the denominator as $(3x+2)(x-2)^2$
Step 2. Assume the end results as $\frac{A}{3x+2}+\frac{B}{x-2}+\frac{C}{(x-2)^2}$
Step 3. Combine the above functions as:
$\frac{A(x-2)^2+B(3x+2)(x-2)+C(3x+2)}{(3x+2)(x-2)^2}=\frac{(A+3B)x^2+(-4A-4B+3C)x+(4A-4B+2C)}{(3x+2)(x-2)^2}$
Step 4. Compare the above result with the original to set up the following system of equations:
\begin{cases} A+3B=3 \\ -4A-4B+3C=5 \\ 4A-4B+2C=-13 \end{cases}
Step 5. Add the last two equations to get $-8B+5C=-8$ which gives $C=\frac{8B-8}{5}$
Step 6 Use $A=3-3B$ together with the above result to back substitute into the last equation and solve for $B$ to get $B=\frac{109}{64}$
Step 7. Get the other parameters as $A=\frac{-135}{64}$ and $C=\frac{9}{8}$
Step 8. Write the end results as:
$-\frac{135}{64(3x+2)}+\frac{109}{64(x-2)}+\frac{9}{8(x-2)^2}$
