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The given rational function is:

\[
t(x) = \frac{(x+2)(x-1)(x-4)}{5(x+3)(x-1)(x-3)}
\]

### Part A: Vertical Asymptotes

To find the vertical asymptotes of the function, we look at the values of \( x \) that make the denominator zero (i.e., where the function is undefined). Vertical asymptotes occur at these points **unless** they are canceled out by a factor in the numerator.

The denominator of the function is:

\[
5(x+3)(x-1)(x-3)
\]

Setting each factor of the denominator equal to zero gives the following values of \( x \):

\[
x + 3 = 0 \implies x = -3
\]
\[
x - 1 = 0 \implies x = 1
\]
\[
x - 3 = 0 \implies x = 3
\]

### Simplifying Common Factors
The numerator also has a factor \( (x-1) \), so the \( (x-1) \) factor cancels out from both the numerator and denominator. Therefore, \( x = 1 \) is **not** a vertical asymptote, but the other factors remain.

### Vertical Asymptotes
The vertical asymptotes are at \( x = -3 \) and \( x = 3 \). Thus, the equations of the vertical asymptotes are:

\[
x = -3 \quad \text{and} \quad x = 3
\]

Would you like more details on this solution or any additional explanations?

### Follow-up Questions:
1. How do you determine the horizontal asymptote of a rational function?
2. What happens to the graph of \( t(x) \) at \( x = 1 \) after the cancellation of the common factor?
3. How can you determine the behavior of the function as \( x \) approaches each vertical asymptote?
4. What are the x-intercepts of this rational function based on its numerator?
5. How would you sketch the graph of this function around the asymptotes and intercepts?

#### Tip:
When factors cancel out, they often lead to a "hole" in the graph rather than an asymptote. Always check for cancellations before determining vertical asymptotes!

Determine equations for all vertical asymptotes of the graph of t(x), or state that no vertical asymptotes exist.

Learn how to find the vertical asymptotes of the rational function t(x) = (x+2)(x-1)(x-4) / [5(x+3)(x-1)(x-3)]. This problem involves analyzing the denominator to determine where it equals zero and whether common factors cancel, leading to vertical asymptotes.

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