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 \( \frac{3 x^{2}+17 x+10}{3 x^{2}+32 x+20} \), we start by factoring both the numerator and the denominator. For the numerator \( 3x^2 + 17x + 10 \), we look for two numbers that multiply to \( 3*10 = 30 \) and add to \( 17 \). Those numbers are \( 15 \) and \( 2 \): \[ 3x^2 + 15x + 2x + 10 = 3x(x + 5) + 2(x + 5) = (3x + 2)(x + 5). \] For the denominator \( 3x^2 + 32x + 20 \), we are looking for two numbers that multiply to \( 3*20 = 60 \) and add to \( 32 \). Those numbers are \( 30 \) and \( 2 \): \[ 3x^2 + 30x + 2x + 20 = 3x(x + 10) + 2(x + 10) = (3x + 2)(x + 10). \] Now, the original expression simplifies to: \[ \frac{(3x + 2)(x + 5)}{(3x + 2)(x + 10)}. \] We can cancel out the common factor \( (3x + 2) \), resulting in: \[ \frac{x + 5}{x + 10}, \quad x \neq -\frac{2}{3}. \] Thus, the simplified expression is: \[ \frac{x + 5}{x + 10}. \]