Sure, I can provide a detailed step-by-step solution for this problem.
Given the rational expressions:
[tex]\[ \frac{x-1}{x+5} \cdot \frac{x+1}{x-5} \][/tex]
To find the product of these rational expressions, follow these steps:
### Step 1: Multiply the numerators together
The numerators are [tex]\(x-1\)[/tex] and [tex]\(x+1\)[/tex].
When you multiply these two binomials, you use the distributive property, specifically the FOIL (First, Outer, Inner, Last) method:
[tex]\[ (x-1)(x+1) \][/tex]
Using the difference of squares formula:
[tex]\[ (a-b)(a+b) = a^2 - b^2 \][/tex]
Here, [tex]\(a = x\)[/tex] and [tex]\(b = 1\)[/tex]. Therefore:
[tex]\[ (x-1)(x+1) = x^2 - 1 \][/tex]
### Step 2: Multiply the denominators together
The denominators are [tex]\(x+5\)[/tex] and [tex]\(x-5\)[/tex].
Similarly, apply the difference of squares formula:
[tex]\[ (x+5)(x-5) = x^2 - 25 \][/tex]
### Step 3: Combine the results
We now have the numerators and denominators multiplied:
[tex]\[ \frac{(x-1)(x+1)}{(x+5)(x-5)} = \frac{x^2 - 1}{x^2 - 25} \][/tex]
### Conclusion
The product of the rational expressions [tex]\(\frac{x-1}{x+5} \cdot \frac{x+1}{x-5}\)[/tex] is [tex]\(\frac{x^2 - 1}{x^2 - 25}\)[/tex].
Hence, the correct option is:
[tex]\[ \boxed{D \; \frac{x^2 - 1}{x^2 - 25}} \][/tex]
