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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.

The rational function f(x) = [(x-3)(x+2)(x-4)^2] / [(x-4)(x+1)]. Which of the following intervals best describes the domain of f?

Learn how to determine the domain of the rational function f(x) = [(x-3)(x+2)(x-4)^2] / [(x-4)(x+1)] by identifying excluded values from the denominator. Excluded values include x = 4 and x = -1. Suitable for high school students, this problem covers key algebra concepts.

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