Step 1: 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. $$ x^2 - x - 2 = (x-2)(x+1) $$ Step 2: Rewrite the expression with the factored denominator. $$ \frac{1}{(x-2)(x+1)} - \frac{1}{x+1} $$ Step 3: Find a common denominator. 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 $(x-2)$. $$ \frac{1}{x+1} = \frac{1 \cdot (x-2)}{(x+1) \cdot (x-2)} = \frac{x-2}{(x+1)(x-2)} $$ Step 4: Subtract the fractions. Now that both fractions have the same denominator, subtract their numerators. $$ \frac{1}{(x-2)(x+1)} - \frac{x-2}{(x-2)(x+1)} = \frac{1 - (x-2)}{(x-2)(x+1)} $$ Step 5: Simplify the numerator. $$ 1 - (x-2) = 1 - x + 2 = 3 - x $$ Step 6: Write the final single fraction. $$ \frac{3 - x}{(x-2)(x+1)} $$ The expression as a single fraction is: $$ \boxed{\frac{3 - x}{(x-2)(x+1)}} $$