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