Answer
$\frac{-3}{x} + \frac{-1}{x+2} + \frac{5}{x-2}$
Work Step by Step
We must first find the Partial Fraction Decomposition:
$\frac{A}{x} + \frac{B}{x+2} + \frac{C}{x-2} = \frac{x^{2}+12x+12}{x^{3}-4x}$
We must then solve for the constants:
$A(x+2)(x-2) + B(x)(x-2) + C(x)(x+2) = x^{2} + 12x + 12$
$Ax^{2}-4A +Bx^{2}-2Bx+Cx^{2}+2Cx = x^{2} + 12x + 12$
$A + B + C = 1$
$-2B + 2C = 12$
$-4A = 12$
This can be represented with the following matrix:
$\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -2 & 2 & |12\\
-4 & 0 & 0 & |12
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -1 & 1 & |6\\
-1 & 0 & 0 & |3
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -1 & 1 & |6\\
0 & 1 & 1 & |4
\end{bmatrix}$ ~ $\begin{bmatrix}
1 & 1 & 1 & |1\\
0 & -1 & 1 & |6\\
0 & 0 & 2 & |10
\end{bmatrix}$
$2C = 10$
$C = 5$
$-B + 5 = 6$
$B = -1$
$A - 1 + 5 = 1$
$A + 4 = 1$
$A = -3$
The Partial Fraction Decomposition is:
$\frac{-3}{x} + \frac{-1}{x+2} + \frac{5}{x-2}$
Checking the Partial Fraction Decomposition:
$-3(x+2)(x-2) - 1(x)(x-2) + 5(x)(x+2)$
$-3(x^{2} - 4) - 1(x^{2}-2x) + 5(x^{2}+2x)$
$-3x^{2}+12-x^{2}+2x+5x^{2}+10x$
$x^{2}+12x+12$
Therefore, the answer is correct.
