Factor the polynomial: \( x^{2}+5 x-14= \)

1. We start with the quadratic polynomial \[ x^2 + 5x - 14. \] 2. We need to find two numbers that multiply to the product of the coefficient of \(x^2\) (which is \(1\)) and the constant term \(-14\), that is, two numbers that multiply to \(-14\), and add up to the coefficient of \(x\), which is \(5\). 3. The two numbers that satisfy these conditions are \(7\) and \(-2\), because \[ 7 \times (-2) = -14 \quad \text{and} \quad 7 + (-2) = 5. \] 4. We now rewrite the middle term \(5x\) as \(7x - 2x\), so: \[ x^2 + 5x - 14 = x^2 + 7x - 2x - 14. \] 5. Next, we factor by grouping: - From the first group \(\,x^2 + 7x\), factor out \(x\): \[ x(x + 7). \] - From the second group \(-2x - 14\), factor out \(-2\): \[ -2(x + 7). \] 6. Now, the expression can be written as: \[ x(x + 7) - 2(x + 7). \] 7. Notice that \((x + 7)\) is a common factor. Factor it out: \[ (x + 7)(x - 2). \] Thus, the factored form of the polynomial is \[ (x + 7)(x - 2). \]