Answer
$\frac{x-2}{x+1}$
Work Step by Step
We switch from division to multiplication by taking the reciprocal of the bottom fraction. Then we factor and simplify:
$\displaystyle\frac{\frac{2x^{2}-3x-2}{x^2-1}}{\frac{2x^{2}+5x+2}{x^2+x-2}}=\frac{2x^{2}-3x-2}{x^{2}-1}*\frac{x^{2}+x-2}{2x^{2}+5x+2}=\frac{(x-2)(2x+1)}{(x-1)(x+1)}*\frac{(x-1)(x+2)}{(x+2)(2x+1)}=\frac{x-2}{x+1}$
