Let's solve the given problem step-by-step to find the quotient of the expression:
[tex]\[
\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}
\][/tex]
### Step 1: Simplify each term individually
First term:
[tex]\[
\frac{3y + 2}{3y}
\][/tex]
This term is already in its simplest form.
Second term:
[tex]\[
\frac{6y^2 + 4y}{3y + 2}
\][/tex]
We can factor out the numerator:
[tex]\[
6y^2 + 4y = 2y(3y + 2)
\][/tex]
So the second term becomes:
[tex]\[
\frac{2y(3y + 2)}{3y + 2}
\][/tex]
We can then cancel [tex]\((3y + 2)\)[/tex] from the numerator and denominator:
[tex]\[
\frac{2y(3y + 2)}{3y + 2} = 2y
\][/tex]
### Step 2: Apply the division
Dividing by a fraction is equivalent to multiplying by its reciprocal:
[tex]\[
\frac{3y + 2}{3y} \div 2y = \frac{3y + 2}{3y} \times \frac{1}{2y}
\][/tex]
### Step 3: Multiply the fractions
To multiply the fractions, multiply the numerators and the denominators:
[tex]\[
\frac{3y + 2}{3y} \times \frac{1}{2y} = \frac{(3y + 2) \times 1}{3y \times 2y} = \frac{3y + 2}{6y^2}
\][/tex]
### Conclusion
The simplified quotient is:
[tex]\[
\frac{3y + 2}{6y^2}
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{\frac{3y+2}{6y^2}}
\][/tex]
