The given function appears to be:

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

Steps for analysis:

1. Denominator factorization: The denominator $x^2 - 3x - 4$ can be factored into: $x^2 - 3x - 4 = (x - 4)(x + 1)$

2. Simplifying the function: Substitute the factored form into the function: $f(x) = \frac{(x - 4)^2}{(x - 4)(x + 1)} - 2$

   For $x \neq 4$, simplify: $f(x) = \frac{x - 4}{x + 1} - 2$

3. Combining terms: Combine into a single fraction: $f(x) = \frac{x - 4 - 2(x + 1)}{x + 1}$

   Expand the numerator: $f(x) = \frac{x - 4 - 2x - 2}{x + 1}$

   Simplify the numerator: $f(x) = \frac{-x - 6}{x + 1}$

Thus, the simplified form of the function is: $f(x) = \frac{-x - 6}{x + 1}, \quad x \neq 4, \, x \neq -1$

Domain:

The function is undefined at $x = 4$ and $x = -1$, where the denominator is zero.

Would you like me to:

1. Explore the asymptotes and behavior of $f(x)$?
2. Plot the function for better visualization?
3. Find specific values or limits?
4. Discuss the behavior near discontinuities ($x = 4$ and $x = -1$)?
5. Verify additional aspects like intercepts?

Tip:

Always check for removable discontinuities after simplifying rational functions.