Simplify: \( \frac{3 x^{2}+17 x+10}{3 x^{2}+32 x+20} \) \( \frac{3 x+2}{3 x+4} \) \( \frac{1}{2} \) \( \frac{x+5}{x+10} \) \( \frac{17 x+10}{32 x+20} \)

To simplify the expression: \[ \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} \] **Step 1: Factor the Numerator and Denominator** 1. **Numerator:** \(3x^{2} + 17x + 10\) - Look for two numbers that multiply to \(3 \times 10 = 30\) and add up to \(17\). - These numbers are \(15\) and \(2\). - Rewrite and factor: \[ 3x^{2} + 15x + 2x + 10 = 3x(x + 5) + 2(x + 5) = (3x + 2)(x + 5) \] 2. **Denominator:** \(3x^{2} + 32x + 20\) - Look for two numbers that multiply to \(3 \times 20 = 60\) and add up to \(32\). - These numbers are \(30\) and \(2\). - Rewrite and factor: \[ 3x^{2} + 30x + 2x + 20 = 3x(x + 10) + 2(x + 10) = (3x + 2)(x + 10) \] **Step 2: Simplify the Expression** \[ \frac{(3x + 2)(x + 5)}{(3x + 2)(x + 10)} = \frac{x + 5}{x + 10} \quad \text{(assuming } 3x + 2 \neq 0\text{)} \] **Final Simplified Form:** \[ \frac{x + 5}{x + 10} \] **Answer:** \[ \frac{x + 5}{x + 10} \]