To factor the quadratic expression \( x^2 + 3x - 40 \), we are looking for two numbers that multiply to \(-40\) (the constant term) and add to \(3\) (the coefficient of the linear term).
The two numbers that meet these criteria are \(8\) and \(-5\), since:
- \(8 \times (-5) = -40\) (they multiply to \(-40\))
- \(8 + (-5) = 3\) (they add up to \(3\))
Using these numbers, we can factor the quadratic as follows:
\[ x^2 + 3x - 40 = (x + 8)(x - 5) \]
Thus, the factors of \( x^2 + 3x - 40 \) are \( (x + 8) \) and \( (x - 5) \).