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 \(x\)).
The numbers that satisfy these conditions are \(-5\) and \(2\), because:
\[ -5 \times 2 = -10 \quad \text{and} \quad -5 + 2 = -3. \]
Now, we can write the expression as:
\[ x^2 - 3x - 10 = (x - 5)(x + 2). \]
Thus, the factored form of \( x^2 - 3x - 10 \) is:
\[ (x - 5)(x + 2). \]