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The given rational function is:

\[
h(x) = \frac{(x + 1)(x - 2)^2}{(x - 1)(x^2 - 4)}
\]

To determine the vertical asymptotes, we need to find the values of \(x\) that make the denominator equal to zero, provided that these values do not also make the numerator zero (which would result in a hole instead of an asymptote).

### Step 1: Factor the denominator
The denominator is \((x - 1)(x^2 - 4)\), where \(x^2 - 4\) is a difference of squares and can be factored as:

\[
x^2 - 4 = (x - 2)(x + 2)
\]

Thus, the denominator becomes:

\[
(x - 1)(x - 2)(x + 2)
\]

### Step 2: Set the denominator equal to zero
Now, we solve:

\[
(x - 1)(x - 2)(x + 2) = 0
\]

This gives three potential vertical asymptote locations at \(x = 1\), \(x = 2\), and \(x = -2\).

### Step 3: Check for holes
A hole occurs if a factor in the denominator is also a factor in the numerator. In the numerator, we have \((x + 1)(x - 2)^2\).

- The factor \(x - 2\) appears in both the numerator and the denominator. However, since \(x - 2\) is squared in the numerator, it cancels only one of the \(x - 2\) factors from the denominator. This means there will be a hole at \(x = 2\).
- The factor \(x + 1\) does not appear in the denominator, so it does not affect any vertical asymptotes.

### Step 4: Conclusion
The vertical asymptotes occur at the values of \(x\) that make the denominator zero but do not correspond to holes. Therefore, the vertical asymptotes are at:

\[
x = 1 \quad \text{and} \quad x = -2
\]

The correct answer is:
\[
\boxed{x = -2 \text{ and } x = 1}
\]

Would you like any further clarification or details? 

Here are 5 related questions to explore:
1. How do you find horizontal asymptotes for rational functions?
2. What is the significance of holes in rational functions?
3. How do you factor more complicated quadratic expressions in rational functions?
4. Can a rational function have both a vertical asymptote and a hole at the same point?
5. How does squaring terms in the numerator affect the behavior of rational functions?

**Tip:** Always check if a factor in the denominator cancels with one in the numerator to determine if there is a hole instead of an asymptote.

A function h(x) is defined as h(x) = [(x + 1)(x - 2)^2] / [(x - 1)(x^2 - 4)]. Determine the location of all vertical asymptotes.

Learn how to find vertical asymptotes of the rational function h(x) = [(x + 1)(x - 2)^2] / [(x - 1)(x^2 - 4)]. This problem covers factoring quadratics and identifying vertical asymptotes while checking for holes. Suitable for high school students in Grades 10-12.

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