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]
Now, update the expression:
[tex]\[
\frac{3y + 2}{3y} \times \frac{3y + 2}{2y(3y + 2)}
\][/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]
