To simplify the expression [tex]\(\frac{4 x^2 + 36}{4 x} \cdot \frac{1}{5 x}\)[/tex], follow these steps:
1. Factor and Simplify the First Fraction:
[tex]\[
\frac{4 x^2 + 36}{4 x}
\][/tex]
Notice that [tex]\(4 x^2 + 36\)[/tex] can be factored:
[tex]\[
4 x^2 + 36 = 4(x^2 + 9)
\][/tex]
So, we have:
[tex]\[
\frac{4(x^2 + 9)}{4 x}
\][/tex]
Simplify the fraction by canceling the common factor of [tex]\(4\)[/tex]:
[tex]\[
\frac{x^2 + 9}{x}
\][/tex]
2. Multiply by the Second Fraction:
Now we need to multiply [tex]\(\frac{x^2 + 9}{x}\)[/tex] by [tex]\(\frac{1}{5 x}\)[/tex]:
[tex]\[
\frac{x^2 + 9}{x} \cdot \frac{1}{5 x}
\][/tex]
3. Combine the Fractions:
When multiplying fractions, multiply the numerators together and the denominators together:
[tex]\[
\frac{(x^2 + 9) \cdot 1}{x \cdot 5 x} = \frac{x^2 + 9}{5 x^2}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{4 x^2 + 36}{4 x} \cdot \frac{1}{5 x}\)[/tex] is:
[tex]\[
\boxed{\frac{x^2 + 9}{5 x^2}}
\][/tex]
Given the multiple choice options, the correct answer is:
C. [tex]\(\frac{x^2 + 9}{5 x^2}\)[/tex]
