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

Let's simplify each of the given fractions: 1. **Simplify** \( \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} \) **Step 1:** Factor the numerator and the denominator. - **Numerator:** \( 3x^{2} + 17x + 10 \) \[ 3x^{2} + 17x + 10 = (3x + 2)(x + 5) \] - **Denominator:** \( 3x^{2} + 32x + 20 \) \[ 3x^{2} + 32x + 20 = (3x + 2)(x + 10) \] **Step 2:** Cancel the common factor \((3x + 2)\). \[ \frac{(3x + 2)(x + 5)}{(3x + 2)(x + 10)} = \frac{x + 5}{x + 10} \] **Simplified Form:** \[ \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} = \frac{x + 5}{x + 10} \] 2. **Simplify** \( \frac{3x + 2}{3x + 4} \) This fraction cannot be simplified further because the numerator and the denominator do not have any common factors other than 1. **Simplified Form:** \[ \frac{3x + 2}{3x + 4} \] 3. **Simplify** \( \frac{1}{2} \) This fraction is already in its simplest form. **Simplified Form:** \[ \frac{1}{2} \] 4. **Simplify** \( \frac{x + 5}{x + 10} \) This fraction cannot be simplified further because the numerator and the denominator do not have any common factors other than 1. **Simplified Form:** \[ \frac{x + 5}{x + 10} \] 5. **Simplify** \( \frac{17x + 10}{32x + 20} \) **Step 1:** Factor the denominator. \[ 32x + 20 = 4(8x + 5) \] **Step 2:** Check if the numerator can be factored further or if there's a common factor with the denominator. The numerator \(17x + 10\) does not have any common factors with the denominator \(8x + 5\) besides 1. **Simplified Form:** \[ \frac{17x + 10}{32x + 20} = \frac{17x + 10}{4(8x + 5)} \] **Summary of Simplified Fractions:** 1. \( \frac{3x^{2} + 17x + 10}{3x^{2} + 32x + 20} = \frac{x + 5}{x + 10} \) 2. \( \frac{3x + 2}{3x + 4} \) 3. \( \frac{1}{2} \) 4. \( \frac{x + 5}{x + 10} \) 5. \( \frac{17x + 10}{32x + 20} = \frac{17x + 10}{4(8x + 5)} \)