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Let's analyze the functions \( f(x) = x^3 - x^2 - 6x \) and \( h(x) = -3(x+1)^2(x+3)^2 \) to match them with their corresponding graphs.

### 1. For \( f(x) = x^3 - x^2 - 6x \):
- This is a cubic function, which typically has an "S" shape.
- The critical points can be found by solving \( f'(x) = 3x^2 - 2x - 6 = 0 \).
- Factoring or solving the derivative helps locate the turning points (local minima and maxima).
- The graph should cross the x-axis at the roots. Setting \( f(x) = 0 \) gives the roots \( x = 0, x = -2, x = 3 \), so the graph will intersect the x-axis at these points.
  
From the given options, **Graph C** has the proper "S" shape with three x-intercepts, so it likely matches \( f(x) \).

### 2. For \( h(x) = -3(x+1)^2(x+3)^2 \):
- This is a quartic function, but the key is the factored form. Both \( (x+1)^2 \) and \( (x+3)^2 \) indicate that the function has double roots at \( x = -1 \) and \( x = -3 \).
- Because the roots are of even multiplicity (squared terms), the graph will touch the x-axis at these points and bounce off, without crossing it.
- Also, the negative leading coefficient \( -3 \) implies the graph opens downward.

From the options, **Graph A** shows a behavior consistent with double roots at \( x = -1 \) and \( x = -3 \), and it opens downward, so this matches \( h(x) \).

### Final Matches:
- \( f(x) = x^3 - x^2 - 6x \) corresponds to **Graph C**.
- \( h(x) = -3(x+1)^2(x+3)^2 \) corresponds to **Graph A**.

Would you like more details or clarifications on any part of this? 

Here are 5 related questions to deepen understanding:
1. How would the graph of \( f(x) \) change if the leading coefficient were negative?
2. What is the impact of changing the multiplicity of roots on the graph's shape?
3. Why does a cubic function like \( f(x) \) always have at least one real root?
4. How do the roots' multiplicity affect the graph at those points?
5. How would the graph of \( h(x) \) change if the coefficient \( -3 \) were positive?

**Tip:** The derivative of a function provides crucial information about the behavior of the graph, such as identifying local minima, maxima, and inflection points.

Choose the graph of each function from the choices below for f(x) = x^3 - x^2 - 6x and h(x) = -3(x+1)^2(x+3)^2.

Match the graph of the cubic function f(x) = x^3 - x^2 - 6x and the quartic function h(x) = -3(x+1)^2(x+3)^2 with the correct visual representations. Learn about roots, multiplicity, and how the sign of the leading coefficient influences the graph's shape.

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