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To solve the inequality \( x^3 + 11x^2 > 0 \), let’s analyze the expression step-by-step.

### Step 1: Factor the Expression
We start by factoring \( x^3 + 11x^2 \):
\[
x^3 + 11x^2 = x^2(x + 11)
\]
So, the inequality becomes:
\[
x^2(x + 11) > 0
\]

### Step 2: Identify the Critical Points
The critical points occur when \( x^2(x + 11) = 0 \), which gives:
1. \( x^2 = 0 \Rightarrow x = 0 \)
2. \( x + 11 = 0 \Rightarrow x = -11 \)

Thus, the critical points are \( x = 0 \) and \( x = -11 \).

### Step 3: Test Intervals Around the Critical Points
We’ll test intervals around these points to determine where the expression is positive:
1. Interval \( (-\infty, -11) \)
2. Interval \( (-11, 0) \)
3. Interval \( (0, \infty) \)

Choose a test point in each interval:

1. **For \( x < -11 \):** (e.g., \( x = -12 \))
   \[
   x^2(x + 11) = (-12)^2(-12 + 11) = 144 \cdot (-1) = -144
   \]
   This is negative.

2. **For \( -11 < x < 0 \):** (e.g., \( x = -1 \))
   \[
   x^2(x + 11) = (-1)^2(-1 + 11) = 1 \cdot 10 = 10
   \]
   This is positive.

3. **For \( x > 0 \):** (e.g., \( x = 1 \))
   \[
   x^2(x + 11) = (1)^2(1 + 11) = 1 \cdot 12 = 12
   \]
   This is positive.

### Step 4: Determine the Solution
From the interval testing, we see that \( x^2(x + 11) > 0 \) in the intervals:
\[
(-11, 0) \cup (0, \infty)
\]

### Step 5: Write the Solution in Interval Notation
The solution in interval notation is:
\[
(-11, 0) \cup (0, \infty)
\]

Would you like further details, or do you have any questions about this solution?

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Here are five related questions you might consider:
1. How would the solution change if the inequality were \( x^3 + 11x^2 \geq 0 \)?
2. What is the graphical representation of \( x^3 + 11x^2 \) on a number line?
3. How would you solve \( x^3 + 11x^2 < 0 \) instead?
4. Why does factoring help in solving polynomial inequalities?
5. What methods can be used to find the critical points in similar inequalities?

**Tip:** Always start by factoring polynomials in inequalities—it helps simplify the problem and identify key points where the sign of the expression changes.

Solve the following inequality: x^3 + 11x^2 > 0

Learn how to solve the polynomial inequality x^3 + 11x^2 > 0 through factoring and interval testing. This detailed step-by-step solution covers critical points, polynomial factorization, and the concept of sign changes. Suitable for students in Grades 9-12.

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