Answer
Solution set: $\{3\}$
Work Step by Step
Factor the denominators
$x^{2}-x=x(x-1)$
$x^{2}-1=(x-1)(x+1)$
LCD = $x(x-1)(x+1)$
$\displaystyle \frac{x+2}{x(x-1)}-\frac{8}{(x-1)(x+1)}=0$
Exclude solutions which yield 0 in denominators:
$x\not\in\{0,-1,1\}\qquad (*)$
Multiply with the LCD, $x(x-1)(x+1)$
$(x+3)(x+1)-8\cdot x=0$
$x^{2}+4x+3-6x=0$
$x^{2}-4x+3=0$
... factors of $3$ whose sum is $-4$ ... are $-1$ and $-3$
$(x-1)(x-3)=0\qquad$ ... apply the zero product principle
$ x=1\qquad$ ... is excluded, see (*).
$x=3$
Solution set: $\{3\}$
