Factor the denominator of the first fraction.

The quadratic expression $x^2 - x - 2$ can be factored into two binomials. We need two numbers that multiply to -2 and add to -1. These numbers are -2 and 1.

$$x^2 - x - 2 = (x-2)(x+1)$$

The common denominator for both fractions is $(x-2)(x+1)$.
The first fraction already has this denominator. For the second fraction, multiply the numerator and denominator by $(x-2)$.

$$\frac{1}{x+1} = \frac{1 \cdot (x-2)}{(x+1) \cdot (x-2)} = \frac{x-2}{(x+1)(x-2)}$$

Now that both fractions have the same denominator, subtract their numerators.

$$\frac{1}{(x-2)(x+1)} - \frac{x-2}{(x-2)(x+1)} = \frac{1 - (x-2)}{(x-2)(x+1)}$$