To factor the quadratic expression \( x^2 - x - 2 \), we need to find two numbers that multiply to \(-2\) (the constant term) and add to \(-1\) (the coefficient of the linear term).
The pair of numbers that meets these criteria is \(-2\) and \(1\) because:
- \(-2 \times 1 = -2\)
- \(-2 + 1 = -1\)
Now we can rewrite the expression using these numbers:
\[ x^2 - x - 2 = x^2 - 2x + x - 2 \]
Next, we can group the terms:
\[ = (x^2 - 2x) + (x - 2) \]
Now, we can factor by grouping:
\[ = x(x - 2) + 1(x - 2) \]
Now factor out the common term \((x - 2)\):
\[ = (x - 2)(x + 1) \]
So the factored form of \( x^2 - x - 2 \) is:
\[ (x - 2)(x + 1) \]