Let's break down the information step by step:

We are given a function:

$P(x) = x(x + 2)(x - 1)$

Step 1: Identify Critical Points

First, we find the roots of $P(x)$, which are the values where the function equals zero:

• $x = 0$
• $x + 2 = 0 \Rightarrow x = -2$
• $x - 1 = 0 \Rightarrow x = 1$

Thus, the critical points where the sign of the function may change are at $x = -2$, $x = 0$, and $x = 1$.

Step 2: Sign Analysis

We now perform a sign analysis on the intervals determined by the critical points: $(- \infty, -2)$, $(-2, 0)$, $(0, 1)$, and $(1, \infty)$.

1. Interval $(- \infty, -2)$:

   • $x < -2$, so $x$ is negative.
   • $x + 2 < 0$, so $x + 2$ is negative.
   • $x - 1 < 0$, so $x - 1$ is negative.
   • The product $x(x + 2)(x - 1)$ is $(-)(-)(-) = -$, so $P(x) < 0$.

2. Interval $(-2, 0)$:

   • $x > -2$ and $x < 0$, so $x$ is negative.
   • $x + 2 > 0$, so $x + 2$ is positive.
   • $x - 1 < 0$, so $x - 1$ is negative.
   • The product $x(x + 2)(x - 1)$ is $(-)(+)(-) = +$, so $P(x) > 0$.

3. Interval $(0, 1)$:

   • $x > 0$ and $x < 1$, so $x$ is positive.
   • $x + 2 > 0$, so $x + 2$ is positive.
   • $x - 1 < 0$, so $x - 1$ is negative.
   • The product $x(x + 2)(x - 1)$ is $(+)(+)(-) = -$, so $P(x) < 0$.

4. Interval $(1, \infty)$:

   • $x > 1$, so $x$ is positive.
   • $x + 2 > 0$, so $x + 2$ is positive.
   • $x - 1 > 0$, so $x - 1$ is positive.
   • The product $x(x + 2)(x - 1)$ is $(+)(+)(+) = +$, so $P(x) > 0$.

Step 3: Analyze at Critical Points

• At $x = -2$, $P(x) = 0$ because of the factor $(x + 2)$.
• At $x = 0$, $P(x) = 0$ because of the factor $x$.
• At $x = 1$, $P(x) = 0$ because of the factor $(x - 1)$.

Step 4: Determine Intervals for $P(x) \geq 0$

From the sign analysis, $P(x) \geq 0$ on the following intervals: $[-2, 0] \cup [1, \infty)$

Thus, the solution in interval notation is: $\boxed{[-2, 0] \cup [1, \infty)}$

Would you like further details or have any questions?

Here are 5 related questions to expand on this topic:

1. What is the significance of the critical points in this analysis?
2. How can we graphically represent the sign changes of $P(x)$?
3. What is the difference between solving for $P(x) \geq 0$ versus $P(x) > 0$?
4. How does the multiplicity of roots affect the behavior of polynomial functions?
5. How can we apply this method to other polynomial functions?

Tip: When analyzing polynomials, the behavior at the roots (whether the function crosses the x-axis or just touches it) depends on the multiplicity of the roots.