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The given function is:

\[
f(x) = \frac{(x - 2)^2 (x + 3)}{(x - 2)(x + 1)}
\]

We need to determine the values of \( x \) for which the function is **not continuous**. To do so, let's analyze the behavior of the function, particularly where the denominator equals zero, as this would cause the function to be undefined.

### Step 1: Analyze the denominator

The denominator is:

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

This equals zero when \( x = 2 \) or \( x = -1 \). These values are points where the function might be undefined, but we need to examine further.

### Step 2: Simplify the function

Notice that there is a common factor of \( (x - 2) \) in both the numerator and the denominator:

\[
f(x) = \frac{(x - 2)^2 (x + 3)}{(x - 2)(x + 1)}
\]

We can cancel one \( (x - 2) \) term:

\[
f(x) = \frac{(x - 2)(x + 3)}{x + 1}
\]

Now, the function is simplified, but we must still consider the domain restrictions from the original form.

- **At \( x = 2 \)**: Even though we canceled a factor, the function was originally undefined at \( x = 2 \). Therefore, there is a **removable discontinuity** at \( x = 2 \).
  
- **At \( x = -1 \)**: The denominator becomes zero at \( x = -1 \), and the function is undefined. This is a **vertical asymptote**, so the function has a discontinuity here.

### Step 3: Check \( x + 3 = 0 \)

From the numerator, \( x + 3 = 0 \) when \( x = -3 \), but this does not cause the function to be undefined, so it does not affect continuity.

### Conclusion

The function is not continuous at:
- \( x = 2 \) (removable discontinuity)
- \( x = -1 \) (vertical asymptote)

Thus, the correct answer is **D) -1 and 2 only**.

---

Would you like further clarification on any part of this solution? Here are some related questions to consider:

1. What is the difference between a removable discontinuity and a vertical asymptote?
2. How do you identify discontinuities in rational functions?
3. Can a function be continuous at a point where the denominator equals zero?
4. What happens to a function at a removable discontinuity?
5. How can we determine if a discontinuity affects the graph of a function?

**Tip**: Always check the domain of a function before simplifying, as simplifying can sometimes hide important restrictions!

Let f be the function given by f(x) = ((x - 2)^2(x + 3)) / ((x - 2)(x + 1)). For which of the following values of x is f not continuous?

Explore how to find discontinuities in the rational function f(x) = ((x - 2)^2(x + 3)) / ((x - 2)(x + 1)), identifying both removable and vertical asymptote discontinuities. Suitable for high school students in Grades 10-12.

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