Decompose the following rational expression into its partial fractions: (5x - 1) / (x² - x - 2)

The denominator is a quadratic expression $x^2 - x - 2$. We need to find two numbers that multiply to $-2$ and add to $-1$. These numbers are 1 and $-2$.

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

Since the denominator has two distinct linear factors, the rational expression can be written as a sum of two fractions with constant numerators.

$$\frac{5x - 1}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}$$

Multiply both sides of the equation by $(x - 2)(x + 1)$:

$$5x - 1 = A(x + 1) + B(x - 2)$$

To find $A$, substitute $x = 2$ into the equation:

$$5(2) - 1 = A(2 + 1) + B(2 - 2)$$ $$10 - 1 = A(3) + B(0)$$ $$9 = 3A$$ $$A = 3$$

To find $B$, substitute $x = -1$ into the equation:

$$5(-1) - 1 = A(-1 + 1) + B(-1 - 2)$$ $$-5 - 1 = A(0) + B(-3)$$ $$-6 = -3B$$ $$B = 2$$

Substitute the values of $A$ and $B$ back into the partial fraction form:

$$\frac{5x - 1}{x^2 - x - 2} = \frac{3}{x - 2} + \frac{2}{x + 1}$$

The partial fraction decomposition is:

$$\frac{3}{x - 2} + \frac{2}{x + 1}$$