The given function is:

$f(x) = \frac{4 - 3x - x^2}{x^2 - 1}$

Let's break it down:

1. Factor the denominator:
   The denominator $x^2 - 1$ is a difference of squares, and can be factored as: $x^2 - 1 = (x - 1)(x + 1)$

2. Rearrange the numerator:
   The numerator $4 - 3x - x^2$ can be rewritten in a more familiar form: $4 - 3x - x^2 = -x^2 - 3x + 4$

Thus, the function becomes:

$f(x) = \frac{-x^2 - 3x + 4}{(x - 1)(x + 1)}$

This is the simplified form of the function. If you'd like, I can help you analyze the function further (e.g., finding intercepts, asymptotes, or limits). Would you like to go into more details?

Here are some related questions:

1. How do you find the vertical asymptotes of a rational function?
2. How can we factor the numerator further to simplify $f(x)$?
3. What is the domain of this rational function?
4. How can we find the intercepts of this function?
5. What is the end behavior of the function as $x \to \infty$?

Tip: Always check if both the numerator and denominator can be factored when simplifying rational functions.