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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?

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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.

Solve the inequality f(x) > 0, where f(x) = x^2(x + 2), by using the graph of the function.

Learn how to solve the polynomial inequality f(x) > 0 for f(x) = x^2(x + 2) by analyzing the graph. This detailed solution includes identifying roots, sign changes, and determining positive intervals, suitable for students in Grades 10-12.

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