To solve the given fraction $\frac{x^2 - 6x + 8}{4 - x^2}$, we need to simplify both the numerator and the denominator by factoring them.

1. Factor the numerator $x^2 - 6x + 8$: $x^2 - 6x + 8 = (x - 2)(x - 4)$

2. Factor the denominator $4 - x^2$: $4 - x^2 = (2 + x)(2 - x) = -(x^2 - 4) = -(x - 2)(x + 2)$

Now, the fraction becomes: $\frac{(x - 2)(x - 4)}{-(x - 2)(x + 2)}$

Next, we can cancel out the common factor $(x - 2)$ in the numerator and the denominator (since $x \neq \pm 2$):

$\frac{(x - 4)}{-(x + 2)} = -\frac{x - 4}{x + 2}$

Therefore, the simplified form of the fraction is: $-\frac{x - 4}{x + 2}$

So, the correct answer is: $\text{B) } -\frac{x - 4}{x + 2}$

If you have any questions or need further details, please let me know. Here are some related questions you might be interested in:

1. How to factor quadratic expressions?
2. How to simplify rational expressions?
3. What are common factors and how to cancel them?
4. How to handle negative signs in rational expressions?
5. How to verify the simplification of algebraic fractions?

Tip: When simplifying algebraic fractions, always look for common factors in the numerator and denominator that can be cancelled out, and be cautious with signs.