We are tasked with solving the inequality:

$(x + 1)(x - 2)^2 \geq 0$

Step 1: Identify critical points

To solve this inequality, we first find the values of $x$ where the expression equals 0. This occurs when either factor is equal to zero.

• $x + 1 = 0 \Rightarrow x = -1$
• $(x - 2)^2 = 0 \Rightarrow x = 2$

Thus, the critical points are $x = -1$ and $x = 2$.

Step 2: Determine the sign of each factor in different intervals

Now, we analyze the expression in the intervals determined by these critical points: $(- \infty, -1)$, $(-1, 2)$, and $(2, \infty)$.

• Interval $(- \infty, -1)$:

  • Choose a test point, say $x = -2$.
  • $(x + 1) = -2 + 1 = -1$
  • $(x - 2)^2 = (-2 - 2)^2 = 16$ (always positive, since it's squared)
  • The product is $(-1) \times 16 = -16$, which is negative.

• Interval $(-1, 2)$:

  • Choose a test point, say $x = 0$.
  • $(x + 1) = 0 + 1 = 1$
  • $(x - 2)^2 = (0 - 2)^2 = 4$
  • The product is $1 \times 4 = 4$, which is positive.

• Interval $(2, \infty)$:

  • Choose a test point, say $x = 3$.
  • $(x + 1) = 3 + 1 = 4$
  • $(x - 2)^2 = (3 - 2)^2 = 1$
  • The product is $4 \times 1 = 4$, which is positive.

Step 3: Check the boundary points

• At $x = -1$, $(x + 1) = 0$ and $(x - 2)^2 = 9$, so the product is $0 \times 9 = 0$, which satisfies the inequality.
• At $x = 2$, $(x + 1) = 3$ and $(x - 2)^2 = 0$, so the product is $3 \times 0 = 0$, which also satisfies the inequality.

Step 4: Combine results

From the analysis of the intervals, we have:

• The expression is negative in $(- \infty, -1)$.
• The expression is positive in $(-1, 2)$ and $(2, \infty)$.
• The expression equals zero at $x = -1$ and $x = 2$.

Thus, the inequality $(x + 1)(x - 2)^2 \geq 0$ is satisfied for:

$x \in [-1, 2] \cup [2, \infty)$

This can be simplified as:

$x \in [-1, \infty)$

Final Answer:

The solution to the inequality is:

$x \in [-1, \infty)$

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Would you like a more detailed explanation or further clarifications on any part of this solution? Here are some related questions you might find interesting:

1. How do we solve quadratic inequalities, such as $x^2 - 5x + 6 \geq 0$?
2. What happens if the inequality is strictly greater than zero (i.e., $(x + 1)(x - 2)^2 > 0$)?
3. How can we use sign charts to solve inequalities involving polynomials?
4. What is the difference between solving an equation and an inequality in terms of boundary points?
5. Can you solve rational inequalities like $\frac{1}{x-2} \geq 0$?

Tip: When solving polynomial inequalities, it's often helpful to first factor the expression, identify critical points, and test the intervals formed by these points.