To determine the domain of the given rational function $f(x)$:

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

we need to identify the values of $x$ that make the denominator equal to zero, as these values will be excluded from the domain. The denominator is:

$(x - 4)(x + 1)$

Setting each factor equal to zero gives the excluded values:

1. $x - 4 = 0$ ⟹ $x = 4$
2. $x + 1 = 0$ ⟹ $x = -1$

These values $x = 4$ and $x = -1$ make the denominator zero and are therefore excluded from the domain. So, the domain of $f(x)$ is all real numbers except $x = 4$ and $x = -1$.

We can express this domain in interval notation. This excludes the points $x = 4$ and $x = -1$:

$(-\infty, -1) \cup (-1, 4) \cup (4, \infty)$

Now, we compare this with the given options:

1. $(-\infty, -2) \cup (-2, 3) \cup (3, \infty)$
2. $(-\infty, -1) \cup (-1, \infty)$
3. $(-\infty, -2) \cup (-2, 3) \cup (3, 4) \cup (4, \infty)$
4. $(-\infty, -1) \cup (-1, 4) \cup (4, \infty)$

The correct answer is:

$(-\infty, -1) \cup (-1, 4) \cup (4, \infty)$

This matches option 4.

Answer:

$(-\infty, -1) \cup (-1, 4) \cup (4, \infty)$

Would you like any more details or have any questions about this problem?

Here are 5 related questions that expand on this information:

1. How do you find the vertical asymptotes of a rational function?
2. What are horizontal asymptotes and how can they be determined for rational functions?
3. How do you simplify a rational function?
4. What is the importance of the factorization in finding the domain of a rational function?
5. How do you determine the points of discontinuity for a rational function?

Tip: Always simplify the rational function as much as possible before determining its domain, as it might help in identifying the excluded values more clearly.