A. [tex][tex]$\frac{1}{2y}$[/tex][/tex] B. [tex][tex]$\frac{3y + 2}{6y^2}$[/tex][/tex] C. [tex][tex]$\frac{1}{y}$[/tex][/tex] D. [tex][tex]$\frac{2(3y + 2)}{3}$[/tex][/tex]
Let's find the quotient of the given expression step by step.
We have to determine the quotient: [tex]\[
\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}
\][/tex]
### Step 1: Rewrite the division as multiplication by the reciprocal We can rewrite the division of fractions as multiplication by the reciprocal of the second fraction. [tex]\[
\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2} = \frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}
\][/tex]
### Step 2: Factor denominators and numerators where possible Let's factor the denominator of the second fraction. [tex]\[
6y^2 + 4y = 2y(3y + 2)
\][/tex]
### Step 3: Simplify by canceling common terms We can cancel the common factor of [tex]\(3y + 2\)[/tex] in the numerator and the denominator of the second fraction: [tex]\[
\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)} = \frac{3y + 2}{3y} \times \frac{1}{2y}
\][/tex] [tex]\[
= \frac{3y + 2}{3y} \times \frac{1}{2y} = \frac{3y + 2}{6y^2}
\][/tex]
Thus, the quotient is: [tex]\[
\frac{3y + 2}{6y^2}
\][/tex]
### Final Answer The correct quotient is: [tex]\[
\boxed{\frac{3y + 2}{6y^2}}
\][/tex]
