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The inequality you provided is:

\[
(x - 1)^2(x + 2)(x - 2) \leq 0
\]

### Step-by-step Solution:

1. **Factorization and Critical Points:**
   - The inequality is already factored as \( (x - 1)^2(x + 2)(x - 2) \leq 0 \).
   - The critical points are values of \(x\) that make each factor zero:
     - \( x = 1 \) (from \( (x - 1)^2 \))
     - \( x = -2 \) (from \( x + 2 \))
     - \( x = 2 \) (from \( x - 2 \))

2. **Behavior at Critical Points:**
   - \( (x - 1)^2 \) is always non-negative because it is squared. At \( x = 1 \), this factor equals zero.
   - The sign of \( (x + 2) \) depends on whether \( x \) is greater or less than -2.
   - The sign of \( (x - 2) \) depends on whether \( x \) is greater or less than 2.

3. **Intervals and Sign Analysis:**
   To determine where the expression is less than or equal to 0, we will examine the signs of each factor on the intervals formed by the critical points:
   - Intervals: \( (-\infty, -2) \), \( (-2, 1) \), \( (1, 2) \), and \( (2, \infty) \).

4. **Sign Table:**
   Let's create a sign table to evaluate the product of the factors:

   | Interval          | \( (x-1)^2 \) | \( x+2 \) | \( x-2 \) | Product \( \leq 0 \)? |
   |-------------------|---------------|-----------|-----------|----------------------|
   | \( (-\infty, -2) \) | +             | -         | -         | +                    |
   | \( (-2, 1) \)       | +             | +         | -         | -                    |
   | \( (1, 2) \)        | +             | +         | -         | +                    |
   | \( (2, \infty) \)   | +             | +         | +         | +                    |

   - For \( x = -2 \): The product equals 0.
   - For \( x = 1 \): The product equals 0.
   - For \( x = 2 \): The product equals 0.

5. **Solution:**
   The inequality \( (x - 1)^2(x + 2)(x - 2) \leq 0 \) holds where the product is less than or equal to zero:
   - The product is negative on the interval \( (-2, 1) \).
   - The product is zero at \( x = -2 \), \( x = 1 \), and \( x = 2 \).

   Therefore, the solution is:

   \[
   x \in [-2, 1] \cup \{2\}
   \]

### Final Answer:
\[
x \in [-2, 1] \cup \{2\}
\]

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Would you like more details or have any questions on this? Here are five related questions to consider:

1. How do you determine the sign of each factor on a number line?
2. What is the significance of repeated roots in inequalities like \( (x - 1)^2 \)?
3. How would the solution change if the inequality were \( > 0 \) instead of \( \leq 0 \)?
4. Why is \( (x - 1)^2 \) always non-negative regardless of \( x \)?
5. How can you verify your solution using a graphing approach?

**Tip:** Always pay attention to whether a factor is squared in inequalities, as this can affect the sign and the behavior at critical points.

Learn how to solve the polynomial inequality (x - 1)^2(x + 2)(x - 2) ≤ 0 with a step-by-step solution. This problem involves factorization, critical points, and sign analysis. Suitable for high school students in Grades 10-12.

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