To determine which of the given options simplifies the rational expression [tex]\(\frac{(x-7)(x+1)}{(x+1)(x-5)}\)[/tex] correctly, we need to follow these steps:
1. Identify and factor the common terms in the numerator and the denominator of the rational expression:
- The numerator is [tex]\((x-7)(x+1)\)[/tex].
- The denominator is [tex]\((x+1)(x-5)\)[/tex].
2. Cancel out the common factors in the numerator and denominator:
- Here, the factor [tex]\((x+1)\)[/tex] is common to both the numerator and denominator.
When [tex]\(x \neq -1\)[/tex], we can safely cancel out the common factor [tex]\((x+1)\)[/tex]:
[tex]\[
\frac{(x-7)(x+1)}{(x+1)(x-5)} = \frac{x-7}{x-5}
\][/tex]
3. Check the simplified expression: Now, we have:
[tex]\[
\frac{x-7}{x-5}
\][/tex]
4. Match the simplified expression with the given options:
- [tex]\( \frac{x-7}{x+1} \)[/tex] does not match our simplified form.
- [tex]\( \frac{x+1}{x-5} \)[/tex] does not match our simplified form.
- [tex]\( \frac{x-7}{x-5} \)[/tex] matches our simplified form.
- [tex]\( \frac{x+1}{x-7} \)[/tex] does not match our simplified form.
So, the correct choice is:
[tex]\(\boxed{\frac{x-7}{x-5}}\)[/tex]
Therefore, the answer is option C.
