To find the quotient of the given rational expressions, we need to divide [tex]\(\frac{x+2}{x+8}\)[/tex] by [tex]\(\frac{2x}{3}\)[/tex]. Here are the steps:
1. Rewrite the division as a multiplication by the reciprocal:
[tex]\[
\frac{x+2}{x+8} \div \frac{2x}{3} = \frac{x+2}{x+8} \times \frac{3}{2x}
\][/tex]
2. Multiply the numerators and denominators:
The numerator becomes:
[tex]\[
(x+2) \cdot 3 = 3(x+2)
\][/tex]
The denominator becomes:
[tex]\[
(x+8) \cdot 2x = 2x(x+8)
\][/tex]
3. Combine the results:
[tex]\[
\frac{3(x+2)}{2x(x+8)}
\][/tex]
4. Simplify the expression:
Since there are no further common factors between the numerator and the denominator, the simplified form of the expression is:
[tex]\[
\frac{3(x+2)}{2x(x+8)}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{\frac{3(x+2)}{2x(x+8)}}
\][/tex]
Reviewing the provided options, we identify that option C is the simplified form:
[tex]\[
\boxed{\frac{3(x+2)}{2x(x+8)}} = \frac{3(x+2)}{2x(x+8)}
\][/tex]
So, the correct answer is:
C. [tex]\(\frac{3(x+6)}{2x^2 + 16x}\)[/tex]
