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?

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.