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To solve the inequality \( f(x) \leq 0 \) using the graph, let's analyze the behavior of the function \( f(x) = -2(x + 2)(x - 3)^3 \).

### Step-by-Step Analysis

1. **Identify the Zeros of \( f(x) \):**
   - The function is zero when \( x + 2 = 0 \) or \( (x - 3)^3 = 0 \).
   - Solving these:
     - \( x + 2 = 0 \Rightarrow x = -2 \)
     - \( (x - 3)^3 = 0 \Rightarrow x = 3 \)

   So, \( f(x) = 0 \) at \( x = -2 \) and \( x = 3 \).

2. **Determine Where \( f(x) \leq 0 \) by Observing the Graph:**
   - The graph crosses the x-axis at \( x = -2 \) and \( x = 3 \).
   - For \( x \) values between \( -2 \) and \( 3 \), the function \( f(x) \) is below the x-axis, indicating \( f(x) < 0 \) in this interval.
   - Outside this interval (i.e., \( x < -2 \) and \( x > 3 \)), the graph shows \( f(x) > 0 \).

3. **Solution in Interval Notation:**
   - Since we are looking for \( f(x) \leq 0 \), we include the points where \( f(x) = 0 \), which are \( x = -2 \) and \( x = 3 \).
   - Therefore, the solution set is:
     \[
     [-2, 3]
     \]

### Final Answer
The solution set for \( f(x) \leq 0 \) is \( \boxed{[-2, 3]} \).

Would you like further details on any step? 

Here are some related questions for further exploration:

1. What are the implications of the cubic term \( (x - 3)^3 \) on the shape of the graph?
2. How does the negative coefficient \( -2 \) affect the direction of the graph?
3. How can we verify the solution by analyzing the derivative of \( f(x) \)?
4. What happens to the solution if the inequality changes to \( f(x) < 0 \)?
5. How would the solution set change if \( f(x) \geq 0 \) were the inequality?

**Tip:** When working with inequalities involving polynomial functions, the zeros of the function help divide the number line into intervals where the function keeps a consistent sign. Analyzing each interval then gives the solution.

Solve the inequality f(x) ≤ 0, where f(x) = -2(x + 2)(x - 3)^3, by using the graph of the function.

Learn to solve the inequality f(x) ≤ 0, where f(x) = -2(x + 2)(x - 3)^3, by analyzing the graph of the function. This problem involves concepts of inequalities, polynomial functions, and graphical interpretation, suitable for students in Grades 10-12.

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