To find which expression is equivalent to [tex]\(\frac{3x}{x+1}\)[/tex] divided by [tex]\(x+1\)[/tex], we need to simplify the division.
The original expression is:
[tex]\[
\frac{\frac{3x}{x+1}}{x+1}
\][/tex]
Now, we can rewrite this expression by multiplying by the reciprocal of [tex]\(x+1\)[/tex]:
[tex]\[
\frac{3x}{x+1} \div x+1 = \frac{3x}{x+1} \cdot \frac{1}{x+1}
\][/tex]
This simplifies our expression to:
[tex]\[
\frac{3x}{(x+1) \cdot (x+1)} = \frac{3x}{(x+1)^2}
\][/tex]
Thus, the equivalent expression to [tex]\(\frac{3x}{x+1}\)[/tex] divided by [tex]\(x+1\)[/tex] is:
[tex]\[
\frac{3x}{x+1} \cdot \frac{1}{x+1}
\][/tex]
Hence, the correct choice is:
[tex]\[
\frac{3x}{x+1} \cdot \frac{1}{x+1}
\][/tex]
