Answer
$\displaystyle \frac{2}{x+2}$
Work Step by Step
To add rational expressions with the same denominator,
add numerators and place the sum over the common denominator.
If possible, factor and simplify the result.
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$\displaystyle \frac{x^{2}-2}{x^{2}+x-2}+\frac{2x-x^{2}}{x^{2}+x-2} = \displaystyle \frac{x^{2}-2+2x-x^{2}}{x^{2}+x-2}$
$= \displaystyle \frac{2x-2}{x^{2}+x-2}$
... to factor the denominator, search for
... two factors of -2 whose sum is +1
... we find +2 and -1
$= \displaystyle \frac{2(x-1)}{(x+2)(x-1)}\qquad$... reduce the common factors
$=\displaystyle \frac{2}{x+2}$