4. \( 3 x^{2}-3 x-16 \) 8. \( 5 x^{2}+35 x+50 \)

Sure, let's factor each of the given quadratic expressions. --- ### **4. \( 3x^{2} - 3x - 16 \)** To factor the quadratic expression \( 3x^{2} - 3x - 16 \), we'll look for two numbers that multiply to \( 3 \times (-16) = -48 \) and add up to \( -3 \). **Step 1:** Identify two numbers whose product is \(-48\) and whose sum is \(-3\). After checking possible pairs, there are no integer pairs that satisfy both conditions. Therefore, the quadratic does not factor neatly over the integers. **Step 2:** Use the **Quadratic Formula** to find the roots: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] For the equation \( 3x^{2} - 3x - 16 = 0 \): - \( a = 3 \) - \( b = -3 \) - \( c = -16 \) \[ x = \frac{-(-3) \pm \sqrt{(-3)^{2} - 4 \times 3 \times (-16)}}{2 \times 3} \] \[ x = \frac{3 \pm \sqrt{9 + 192}}{6} \] \[ x = \frac{3 \pm \sqrt{201}}{6} \] **Conclusion:** The expression does not factor neatly over integers and its roots are: \[ x = \frac{3 + \sqrt{201}}{6} \quad \text{and} \quad x = \frac{3 - \sqrt{201}}{6} \] --- ### **8. \( 5x^{2} + 35x + 50 \)** To factor the quadratic expression \( 5x^{2} + 35x + 50 \), follow these steps: **Step 1:** Factor out the greatest common factor (GCF) from all terms. \[ 5x^{2} + 35x + 50 = 5(x^{2} + 7x + 10) \] **Step 2:** Factor the quadratic inside the parentheses. We need two numbers that multiply to \( 10 \) and add up to \( 7 \). - These numbers are \( 2 \) and \( 5 \). \[ x^{2} + 7x + 10 = (x + 2)(x + 5) \] **Final Factored Form:** \[ 5x^{2} + 35x + 50 = 5(x + 2)(x + 5) \] --- **Summary:** - **Problem 4:** The quadratic \( 3x^{2} - 3x - 16 \) does not factor neatly over the integers. Its roots are \( x = \frac{3 \pm \sqrt{201}}{6} \). - **Problem 8:** The quadratic \( 5x^{2} + 35x + 50 \) factors to \( 5(x + 2)(x + 5) \).