Alright, let's simplify the given expression step-by-step.
Given expression:
[tex]\[
\frac{2}{x^2-25} - \frac{x+3}{x-5}
\][/tex]
First, let's factor the denominator [tex]\(x^2-25\)[/tex]:
[tex]\[
x^2-25 = (x-5)(x+5)
\][/tex]
So, the expression becomes:
[tex]\[
\frac{2}{(x-5)(x+5)} - \frac{x+3}{x-5}
\][/tex]
To combine these fractions, we need a common denominator. The common denominator is [tex]\((x-5)(x+5)\)[/tex]. Therefore, we rewrite the second fraction with this common denominator:
[tex]\[
\frac{2}{(x-5)(x+5)} - \frac{(x+3)(x+5)}{(x-5)(x+5)}
\][/tex]
Now, combine both fractions into a single expression:
[tex]\[
\frac{2 - (x+3)(x+5)}{(x-5)(x+5)}
\][/tex]
Expand the numerator:
[tex]\[
\frac{2 - (x^2 + 5x + 3x + 15)}{(x-5)(x+5)}
\][/tex]
Combine like terms in the numerator:
[tex]\[
\frac{2 - (x^2 + 8x + 15)}{(x-5)(x+5)}
\][/tex]
Distribute the negative sign inside the numerator:
[tex]\[
\frac{2 - x^2 - 8x - 15}{(x-5)(x+5)}
\][/tex]
Combine all the terms in the numerator:
[tex]\[
\frac{-x^2 - 8x - 13}{(x-5)(x+5)}
\][/tex]
Which simplifies to:
[tex]\[
\frac{-x^2 - 8x - 13}{x^2 - 25}
\][/tex]
From the given choices, this corresponds to:
C. [tex]\(\frac{-x^2 - 8 x - 13}{x^2-25}\)[/tex]
So, the correct answer is:
[tex]\[
\boxed{C}
\][/tex]
