The images you uploaded show a polynomial graph $p(x)$ and a multiple-choice question asking to identify the correct polynomial expression based on the graph.

Analysis of the Graph:

The graph is a polynomial with the following key features:

1. Degree: The graph has 3 turning points, indicating that the polynomial is at least of degree 4.
2. Behavior at Extremes: The graph goes to $+\infty$ as $x$ approaches $-\infty$, and also to $+\infty$ as $x$ approaches $+\infty$. This indicates that the leading term has an even degree with a positive coefficient.
3. Roots: The graph crosses the x-axis at three points, indicating three real roots. The exact roots appear to be around $x = -3, 0, 2$.

Provided Options:

The provided options are:

• $p(x) = x^3 - 9x + 1$
• $p(x) = 2x^4 - x^3$
• $p(x) = x^4 - 5x^2 + 4$ (Selected in the image)
• $p(x) = 5x^4 - 4x^3 - 9$

Matching the Graph with the Correct Polynomial:

• $p(x) = x^4 - 5x^2 + 4$ seems to be the correct choice:
  • It is a quartic polynomial (degree 4), matching the number of turning points and the end behavior.
  • The roots at $x \approx -3, 0, 2$ match the graph.

Thus, the selected answer $p(x) = x^4 - 5x^2 + 4$ appears to be correct.

Would you like further details on the analysis or have any other questions?

Here are some related questions for further exploration:

1. What are the steps to determine the degree of a polynomial from its graph?
2. How do turning points relate to the degree of a polynomial?
3. What is the significance of the leading coefficient in a polynomial function?
4. How can you determine the roots of a polynomial function by examining its graph?
5. What are the different types of polynomial roots, and how do they appear on a graph?

Tip: To determine the degree of a polynomial from its graph, count the number of turning points and remember that a polynomial of degree $n$ can have at most $n-1$ turning points.