Sure, let's determine the product of the given rational expressions step-by-step and simplify the result.
We have the product of two rational expressions:
[tex]\[
\frac{x+1}{x-4} \cdot \frac{5 x}{x+1}
\][/tex]
### Step-by-Step Solution:
1. Write down the given expressions:
[tex]\[
\frac{x+1}{x-4} \cdot \frac{5 x}{x+1}
\][/tex]
2. Multiply the numerators together:
The numerators are [tex]\((x + 1)\)[/tex] and [tex]\(5x\)[/tex]. Multiplying these together, we get:
[tex]\[
(x + 1) \cdot 5x = 5x \cdot (x + 1)
\][/tex]
3. Multiply the denominators together:
The denominators are [tex]\((x - 4)\)[/tex] and [tex]\((x + 1)\)[/tex]. Multiplying these together, we get:
[tex]\[
(x - 4) \cdot (x + 1)
\][/tex]
4. Construct the new fraction with the multiplied numerators and denominators:
[tex]\[
\frac{5x \cdot (x + 1)}{(x - 4) \cdot (x + 1)}
\][/tex]
5. Simplify the fraction by canceling common factors in the numerator and the denominator:
The factor [tex]\((x + 1)\)[/tex] is present in both the numerator and the denominator. So we can cancel them out:
[tex]\[
\frac{5x \cdot \cancel{(x + 1)}}{(x - 4) \cdot \cancel{(x + 1)}} = \frac{5x}{x - 4}
\][/tex]
6. Write the simplified expression:
[tex]\[
\frac{5x}{x - 4}
\][/tex]
### Final Result:
The simplified product of the rational expressions is:
[tex]\[
\boxed{\frac{5 x}{x-4}}
\][/tex]
So, the correct choice is [tex]\(\boxed{D}\)[/tex].
