I am almost certain you mean
x/(x-2) + (x-1)/(x+1) = -1
the LCD is (x-2)(x+1) , so multiply each term by that to get
x(x+1) + (x-1)(x-2) = -1(x-2)(x+1)
expand, re-arrange to get a quadratic.
Then solve using your favourite method.
wait what do you mean your favorite method?
Methods to solve quadratics:
1. by factoring, will not always work
2. by completing the square, best way if coefficient of the square term is 1 and the middle term is even
3. using the quadratic formula, sure-proof way