Certainly! Let's carefully go through the process of finding the quotient step-by-step.
Given the expressions:
[tex]\[ \text{Expression 1: } \frac{3y + 2}{3y} \][/tex]
[tex]\[ \text{Expression 2: } \frac{6y^2 + 4y}{3y + 2} \][/tex]
We need to find the quotient:
[tex]\[ \frac{\frac{3y + 2}{3y}}{\frac{6y^2 + 4y}{3y + 2}} \][/tex]
To divide the two fractions, we multiply the first fraction by the reciprocal of the second fraction:
[tex]\[ \frac{3y + 2}{3y} \times \frac{3y + 2}{6y^2 + 4y} \][/tex]
Now, we simplify this step-by-step:
1. Multiply the numerators:
[tex]\[ (3y + 2) \times (3y + 2) = (3y + 2)^2 \][/tex]
2. Multiply the denominators:
[tex]\[ 3y \times (6y^2 + 4y) = 3y \times (2y \cdot 3y + 4y) = 3y \times (6y^2 + 4y) \][/tex]
3. Rewrite our fraction:
[tex]\[ \frac{(3y + 2)^2}{3y \times (6y^2 + 4y)} \][/tex]
Since [tex]\( (6y^2 + 4y) = 2y(3y + 2) \)[/tex], we substitute to obtain:
[tex]\[ \frac{(3y + 2)^2}{3y \times 2y(3y + 2)} = \frac{(3y + 2)^2}{6y^2 (3y + 2)} \][/tex]
Next, we cancel one [tex]\( (3y + 2) \)[/tex] term from the numerator and the denominator:
[tex]\[ \frac{3y + 2}{6y^2} \][/tex]
Thus, the simplified quotient is:
[tex]\[ \frac{3y + 2}{6y^2} \][/tex]
So, the correct answer is:
[tex]\[ \boxed{\frac{3y + 2}{6y^2}} \][/tex]
