Answer
$\displaystyle \frac{2(5x-1)}{(x-2)(x+1)^{2}}$
Work Step by Step
$\displaystyle \frac{\frac{x+4}{x-2}-\frac{x-3}{x+1}}{x+1}=(\frac{x+4}{x-2}-\frac{x-3}{x+1})\div(x+1)$
$=(\displaystyle \frac{x+4}{x-2}\cdot\frac{x+1}{x+1}-\frac{x-3}{x+1}\cdot\frac{x-2}{x-2})\div(x+1)$
$=(\displaystyle \frac{x^{2}+5x+4}{(x-2)(x+1)}-\frac{x^{2}-5x+6}{(x-2)(x+1)})\div(x+1)$
$=(\displaystyle \frac{10x-2}{(x-2)(x+1)})\div(x+1)$
... division = multiplication with the reciprocal,
$=\displaystyle \frac{2(5x-1)}{(x-2)(x+1)}\cdot\frac{1}{(x+1)}$
$=\displaystyle \frac{2(5x-1)}{(x-2)(x+1)^{2}}$
