What is the quotient? \( \frac{3 y+2}{3 y} \div \frac{6 y^{2}+4 y}{3 y+2} \)

To solve the expression \( \frac{3 y+2}{3 y} \div \frac{6 y^{2}+4 y}{3 y+2} \), we can first rewrite the division as multiplication by the reciprocal: \[ \frac{3 y+2}{3 y} \times \frac{3 y + 2}{6 y^{2} + 4 y} \] Now, we can simplify \( 6 y^{2} + 4 y \) in the denominator. Factoring out the greatest common factor gives: \[ 6 y^{2} + 4 y = 2y(3y + 2) \] So the expression becomes: \[ \frac{3 y + 2}{3 y} \times \frac{3 y + 2}{2y(3 y + 2)} \] Notice that \( 3 y + 2 \) in the numerator and the denominator can be canceled out (provided \( 3 y + 2 \neq 0 \)): \[ \frac{1}{3 y} \times \frac{3 y + 2}{2y} \] Next, we simplify it: \[ \frac{3 y + 2}{6 y^2} \] This is the simplified form of the original expression. Thus, the quotient is: \[ \frac{3y + 2}{6y^2} \]