Factor \( 5 x^{2}-x-4 \)

To factor the quadratic expression \(5x^{2} - x - 4\), follow these steps: 1. **Identify the Coefficients:** - \(a = 5\) (coefficient of \(x^{2}\)) - \(b = -1\) (coefficient of \(x\)) - \(c = -4\) (constant term) 2. **Find Two Numbers that Multiply to \(a \times c = 5 \times (-4) = -20\) and Add to \(b = -1\):** - The numbers \(-5\) and \(4\) satisfy these conditions since: \[ -5 \times 4 = -20 \quad \text{and} \quad -5 + 4 = -1 \] 3. **Rewrite the Middle Term Using These Numbers:** \[ 5x^{2} - 5x + 4x - 4 \] 4. **Factor by Grouping:** - Group the terms: \[ (5x^{2} - 5x) + (4x - 4) \] - Factor out the common factors in each group: \[ 5x(x - 1) + 4(x - 1) \] - Now, factor out the common binomial \((x - 1)\): \[ (5x + 4)(x - 1) \] 5. **Final Factored Form:** \[ 5x^{2} - x - 4 = (5x + 4)(x - 1) \] **Verification:** Multiply the factored form to ensure it equals the original expression: \[ (5x + 4)(x - 1) = 5x \cdot x + 5x \cdot (-1) + 4 \cdot x + 4 \cdot (-1) = 5x^{2} - 5x + 4x - 4 = 5x^{2} - x - 4 \] This confirms the factoring is correct. **Answer:** \[ 5x^{2} - x - 4 = (5x + 4)(x - 1) \]