To combine the given rational expressions, we first need to express them with a common denominator. Let's work through this step-by-step.
We are given the following expressions:
\[ -\frac{4}{x-4} + \frac{x^2 - 2}{2x - 8} \]
Notice that \(2x - 8\) can be factored as \(2(x - 4)\). So we will rewrite the second expression:
\[ -\frac{4}{x-4} + \frac{x^2 - 2}{2(x - 4)} \]
Now, we can find a common denominator, which will be \(2(x - 4)\).
We rewrite the first expression:
\[ -\frac{4}{x-4} = -\frac{4 \cdot 2}{(x-4) \cdot 2} = -\frac{8}{2(x-4)} \]
Now we can combine both fractions:
\[ -\frac{8}{2(x - 4)} + \frac{x^2 - 2}{2(x - 4)} = \frac{-8 + (x^2 - 2)}{2(x - 4)} \]
Now, simplify the numerator:
\[ -8 + x^2 - 2 = x^2 - 10 \]
Thus, we have:
\[ \frac{x^2 - 10}{2(x - 4)} \]
At this point, we cannot simplify \(x^2 - 10\) further, so the final simplified expression is:
\[ \frac{x^2 - 10}{2(x - 4)} \]
Among your provided answer choices, this corresponds to none exactly. The closest choice with a similar numerator would be:
\[ \frac{x^2 - 2x - 8}{2x - 8} \]
However, since our final expression is already simplified, the correct answer for the sum of these rational expressions in simplest form is:
\[ \frac{x^2 - 10}{2(x - 4)} \]