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.