Answer
$\frac{-x^{2}-2x+3}{(x+2)(x-2)}; x \ne 2,-2;$
Work Step by Step
$\frac{x+5}{x^{2}-4} - \frac{x+1}{x-2}$
Factorize $(x^{2}-4) = (x^{2}- 2^{2}) = (x+2)(x-2)$
$[(a^{2}- b^{2})=(a+b)(a-b)]$
$= \frac{x+5}{(x+2)(x-2)} - \frac{x+1}{(x-2)}; x \ne 2,-2;$
Take LCD,
$= \frac{x+5-(x+1)(x+2)}{(x+2)(x-2)}; x \ne 2,-2;$
$= \frac{x+5-(x^{2}+2x+x+2)}{(x+2)(x-2)}; x \ne 2,-2;$
$= \frac{x+5-(x^{2}+3x+2)}{(x+2)(x-2)}; x \ne 2,-2;$
$= \frac{x+5-x^{2}-3x-2}{(x+2)(x-2)}; x \ne 2,-2;$
$= \frac{-x^{2}-2x+3}{(x+2)(x-2)}; x \ne 2,-2;$
