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!