Answer
$\frac{3}{x-1}-\frac{2}{x^2+2}$
Work Step by Step
Step 1. Factor the denominator as: $x^3+2x-(x^2+2)=(x^2+2)(x-1)$
Step 2. Assume the end results as: $\frac{A}{x-1}+\frac{Bx+C}{x^2+2}$
Step 3. Combine the above functions as:
$\frac{A(x^2+2)+(Bx+C)(x-1)}{(x^2+2)(x-1)}=\frac{(A+B)x^2+(-B+C)x+(2A-C)}{(x^2+2)(x-1)}$
Step 4. Compare the above result with the original expression to set up the following system of equations:
\begin{cases} A+B=3 \\ -B+C=-2\\2A-C=8 \end{cases}
Step 5. Add up all equations to get $3A=9$ or $A=3$, use substitution to get $B=0, C=-2$
Step 6. Write the final results as:
$\frac{3}{x-1}-\frac{2}{x^2+2}$
