https://file.aiquickdraw.com/mathbot/file/1726777376068wbdndmh2.jpg

We are tasked with finding a polynomial function \( p(x) \) that satisfies several given conditions. Let's analyze the conditions one by one:

1. **As \( x \to -\infty \), \( p(x) \to \infty \)**, and as \( x \to \infty \), \( p(x) \to -\infty \)**.
    - This indicates that the polynomial has odd degree (likely cubic, since it switches signs at infinity).

2. **The graph touches the x-axis at \( (-4, 0) \)**.
    - This suggests a root at \( x = -4 \), and because the graph "touches" the x-axis, the root is a repeated one (specifically a double root).

3. **\( p(0) = 4 \)**.
    - This gives us a specific point that the function passes through: \( (0, 4) \).

4. **\( p(3) = 0 \)**.
    - This is another root of the polynomial, meaning \( x = 3 \) is a single root.

### General Form of the Polynomial:
Based on these conditions, we can propose a general form for the polynomial:

\[
p(x) = a(x + 4)^2(x - 3)
\]

- \( (x + 4)^2 \) corresponds to the double root at \( x = -4 \).
- \( (x - 3) \) corresponds to the single root at \( x = 3 \).

### Use \( p(0) = 4 \) to Find \( a \):
Substituting \( x = 0 \) into the polynomial:

\[
p(0) = a(0 + 4)^2(0 - 3) = 4
\]

\[
a(16)(-3) = 4
\]

\[
-48a = 4
\]

\[
a = -\frac{1}{12}
\]

Thus, the polynomial function is:

\[
p(x) = -\frac{1}{12}(x + 4)^2(x - 3)
\]

This satisfies all the given conditions, including the behavior as \( x \to \infty \) and \( x \to -\infty \), the roots, and the point \( (0, 4) \).

### Conclusion:
The correct polynomial function is:

\[
p(x) = -\frac{1}{12}(x + 4)^2(x - 3)
\]

The expression in the image was likely incorrect because of the wrong constant factor.

Would you like further clarification on any of the steps? Here are five related questions to explore further:

1. How do we determine the degree of a polynomial from its behavior as \( x \to \infty \)?
2. Why does a double root at \( x = -4 \) result in the graph touching the x-axis instead of crossing it?
3. What would the polynomial look like if the graph passed through \( (-4, 0) \) instead of touching it?
4. How do we find \( a \) in a polynomial function when given a point?
5. Can you explain the significance of each factor in the polynomial \( p(x) = -\frac{1}{12}(x + 4)^2(x - 3) \)?

**Tip:** When analyzing polynomial roots, the multiplicity (how many times a root is repeated) can determine whether the graph crosses or merely touches the x-axis at that point.

Find a polynomial function p(x) that satisfies the following criteria:
- As x → -∞, p(x) → ∞, and as x → ∞, p(x) → -∞.
- The graph of y = p(x) touches the x-axis at (-4, 0).
- p(0) = 4.
- p(3) = 0.
- As x → ∞, p(x) → -∞.

Learn how to find a polynomial function that satisfies conditions such as touching the x-axis at a root, specific values at points, and behavior at infinity. This problem covers concepts of polynomial roots and multiplicity, along with end behavior. Suitable for Grades 11-12.

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