https://file.aiquickdraw.com/mathbot/file/1730944170760od3wv2ps.jpg

To solve the inequality given in the image:

\[
x^2(x + 4)(x - 1) > 0
\]

we need to analyze the inequality by finding the intervals where this expression is positive.

### Step 1: Identify the Zeros
The inequality is a product of three terms. We start by finding the zeros of each term:
1. \( x^2 = 0 \) implies \( x = 0 \)
2. \( x + 4 = 0 \) implies \( x = -4 \)
3. \( x - 1 = 0 \) implies \( x = 1 \)

These zeros are \( x = 0 \), \( x = -4 \), and \( x = 1 \).

### Step 2: Analyze Intervals
Using these zeros, we divide the real line into intervals:
1. \( (-\infty, -4) \)
2. \( (-4, 0) \)
3. \( (0, 1) \)
4. \( (1, \infty) \)

For each interval, we test a point within the interval to see if the expression \( x^2(x + 4)(x - 1) \) is positive or negative.

### Step 3: Sign Analysis
1. **Interval \( (-\infty, -4) \):** Pick \( x = -5 \).
   \[
   x^2 = 25 \quad (positive), \quad (x+4) = -1 \quad (negative), \quad (x-1) = -6 \quad (negative)
   \]
   The product \( 25 \times (-1) \times (-6) = 150 \) (positive).

2. **Interval \( (-4, 0) \):** Pick \( x = -1 \).
   \[
   x^2 = 1 \quad (positive), \quad (x+4) = 3 \quad (positive), \quad (x-1) = -2 \quad (negative)
   \]
   The product \( 1 \times 3 \times (-2) = -6 \) (negative).

3. **Interval \( (0, 1) \):** Pick \( x = 0.5 \).
   \[
   x^2 = 0.25 \quad (positive), \quad (x+4) = 4.5 \quad (positive), \quad (x-1) = -0.5 \quad (negative)
   \]
   The product \( 0.25 \times 4.5 \times (-0.5) = -0.5625 \) (negative).

4. **Interval \( (1, \infty) \):** Pick \( x = 2 \).
   \[
   x^2 = 4 \quad (positive), \quad (x+4) = 6 \quad (positive), \quad (x-1) = 1 \quad (positive)
   \]
   The product \( 4 \times 6 \times 1 = 24 \) (positive).

### Step 4: Solution
The intervals where \( x^2(x + 4)(x - 1) > 0 \) (positive) are:
- \( (-\infty, -4) \)
- \( (1, \infty) \)

Thus, the solution to the inequality is:
\[
x \in (-\infty, -4) \cup (1, \infty)
\]

Let me know if you need further details or explanations on any of the steps.

Here are five related questions for further exploration:
1. How would the solution change if the inequality were \( \geq 0 \) instead of \( > 0 \)?
2. What would the solution be if the inequality were \( x^2(x + 4)(x - 1) < 0 \)?
3. Can this polynomial inequality be solved graphically, and how?
4. How do changes to the coefficients affect the intervals where the inequality holds?
5. How can interval notation help represent the solution clearly in complex inequalities?

**Tip:** For polynomial inequalities, always find the zeros first, then use interval testing to determine the sign of the expression within each interval.

Consider the following polynomial inequality: x^2(x + 4)(x - 1) > 0. Write the polynomial inequality in the form p(x) < 0, p(x) ≤ 0, p(x) > 0, or p(x) ≥ 0, then find the real zeros of p(x).

Learn how to solve the polynomial inequality x^2(x + 4)(x - 1) > 0 using interval testing and sign analysis. This step-by-step solution finds real zeros and analyzes intervals, suitable for high school students in Grades 10-12.

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