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:

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

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.