To solve the inequality $f(x) > 0$ for the function $f(x) = x^2(x + 2)$ using the graph, we need to identify the intervals where the function's values are positive.

1. Identify the roots:

   • From the graph, it looks like $f(x) = 0$ at $x = -2$ and $x = 0$ (the points where the curve crosses the x-axis).

2. Analyze the sign of $f(x)$ in intervals:

   • For $x < -2$, the graph is below the x-axis, so $f(x) < 0$.
   • Between $x = -2$ and $x = 0$, the graph is above the x-axis, meaning $f(x) > 0$.
   • For $x > 0$, the graph is also above the x-axis, so $f(x) > 0$.

3. Solution in interval notation:

   • The function is positive for $x \in (-2, 0) \cup (0, \infty)$.

So, the solution set for $f(x) > 0$ is: $(-2, 0) \cup (0, \infty)$

Would you like more details on how the function behaves in each interval or have any other questions?

Here are some related questions to deepen understanding:

1. How would the solution change if the inequality was $f(x) \geq 0$?
2. What is the effect of changing $f(x)$ to $f(x) = x^2(x + 2)^2$?
3. How do we interpret the multiplicity of roots when analyzing the graph of a polynomial?
4. Can you identify intervals where $f(x) < 0$ based on this graph?
5. How would the graph change if we added a constant term to $f(x)$, like $f(x) = x^2(x + 2) + 1$?

Tip: When finding intervals for polynomial inequalities, check the roots and analyze the sign of the function in each interval between the roots.