To solve the problem of finding the product of the rational expressions [tex]\(\frac{2}{x+1} \cdot \frac{5}{3x}\)[/tex], we'll follow several steps:
1. Numerator Multiplication:
Multiply the numerators of the given rational expressions:
[tex]\[
\text{Numerator} = 2 \cdot 5 = 10
\][/tex]
2. Denominator Multiplication:
Multiply the denominators of the given rational expressions:
[tex]\[
\text{Denominator} = (x + 1) \cdot (3x)
\][/tex]
3. Expand the Denominator:
Distribute [tex]\(3x\)[/tex] into [tex]\((x + 1)\)[/tex]:
[tex]\[
(x + 1) \cdot 3x = 3x \cdot x + 3x \cdot 1 = 3x^2 + 3x
\][/tex]
4. Form the Resulting Rational Expression:
Combine the results for the numerator and the denominator:
[tex]\[
\frac{10}{3x^2 + 3x}
\][/tex]
Thus, the product of [tex]\(\frac{2}{x+1} \cdot \frac{5}{3x}\)[/tex] is [tex]\(\frac{10}{3x^2 + 3x}\)[/tex].
Therefore, the correct answer is:
C. [tex]\(\frac{10}{3x^2 + 3x}\)[/tex]
