Sure, let's solve the problem step-by-step to find the correct quotient.
Given the expression:
[tex]\[
\frac{3y + 2}{3y} \div \frac{6y^2 + 4y}{3y + 2}
\][/tex]
First, recall that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we rewrite the expression as:
[tex]\[
\frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y}
\][/tex]
Now, multiply the numerators and the denominators:
[tex]\[
\frac{(3y + 2) \times (3y + 2)}{3y \times (6y^2 + 4y)}
\][/tex]
Next, simplify the numerator:
[tex]\[
(3y + 2) \times (3y + 2) = (3y + 2)^2
\][/tex]
So, this becomes:
[tex]\[
\frac{(3y + 2)^2}{3y \times (6y^2 + 4y)}
\][/tex]
Move to the denominator now; factor out a common factor in the term [tex]\(6y^2 + 4y\)[/tex]:
[tex]\[
6y^2 + 4y = 2y(3y + 2)
\][/tex]
Replacing this in the expression:
[tex]\[
\frac{(3y + 2)^2}{3y \times 2y (3y + 2)}
\][/tex]
Cancel the common factor of [tex]\(3y + 2\)[/tex] from the numerator and denominator:
[tex]\[
\frac{(3y + 2)}{3y \times 2y} = \frac{(3y + 2)}{6y^2}
\][/tex]
Thus, the quotient simplifies to:
[tex]\[
\frac{3y + 2}{6y^2}
\][/tex]
Therefore, the answer is:
[tex]\[
\boxed{\frac{3 y + 2}{6 y^2}}
\][/tex]
