To factor the quadratic expression \(x^2 - 3x - 10\), we need to find two numbers that multiply to \(-10\) (the constant term) and add up to \(-3\) (the coefficient of the linear term).
The two numbers that satisfy these conditions are \(2\) and \(-5\), because:
- \(2 \times (-5) = -10\) (they multiply to \(-10\))
- \(2 + (-5) = -3\) (they add up to \(-3\))
Thus, we can factor the expression as follows:
\[ x^2 - 3x - 10 = (x + 2)(x - 5) \]
So, the factored form of the quadratic expression \(x^2 - 3x - 10\) is \((x + 2)(x - 5)\).