Let's analyze the problem:

The given polynomial functions are:

1. $g(x) = (x - 4)^2 (x^2 - 4)$
2. $h(x) = -4x^4 + 12x^3$

We need to match these functions to the appropriate graphs. Here's the process:

Step 1: Analyze $g(x) = (x - 4)^2 (x^2 - 4)$

• Factorization: $g(x) = (x - 4)^2 (x - 2)(x + 2)$
• Roots: $x = 4$ (double root), $x = 2$, and $x = -2$.
• Behavior at roots:
  • At $x = 4$: The graph touches the x-axis (since it's a double root).
  • At $x = 2$ and $x = -2$: The graph crosses the x-axis (single roots).
• End behavior: $g(x) \sim x^4$, so as $x \to \pm\infty$, $g(x) \to \infty$.

Step 2: Analyze $h(x) = -4x^4 + 12x^3$

• Factorization: $h(x) = -4x^3(x - 3)$
• Roots: $x = 0$ (triple root) and $x = 3$ (single root).
• Behavior at roots:
  • At $x = 0$: The graph flattens and crosses the x-axis (triple root).
  • At $x = 3$: The graph crosses the x-axis (single root).
• End behavior: $h(x) \sim -4x^4$, so as $x \to \infty$, $h(x) \to -\infty$, and as $x \to -\infty$, $h(x) \to -\infty$.

Step 3: Match to the Graphs

1. Graph for $g(x)$:

   • Look for a graph with roots at $x = -2, 2, 4$ and end behavior $\to \infty$ on both sides.
   • This matches Graph C.

2. Graph for $h(x)$:

   • Look for a graph with roots at $x = 0, 3$ and end behavior $\to -\infty$ on both sides.
   • This matches Graph F.

Final Answer:

• $g(x)$: Graph C
• $h(x)$: Graph F

Would you like further explanation or clarification on these matches?

Related Questions:

1. What does it mean for a root to be "double" or "triple" in polynomial behavior?
2. How do you determine end behavior from the leading term of a polynomial?
3. Why does $g(x)$ touch the x-axis at $x = 4$?
4. What happens to a graph at a "flattening" point like a triple root?
5. How can symmetry (even/odd functions) help identify polynomial graphs?

Tip:

Always check the multiplicity of roots and end behavior to quickly narrow down graph choices.