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The problem involves identifying the equation of a polynomial \( p(x) \) from its graph. Let’s analyze the graph and options:

1. **Intercepts and roots**:
   - The graph touches or crosses the x-axis at \( x = -4 \), \( x = -1 \), and \( x = 1 \).
   - At \( x = -4 \), the graph crosses the x-axis, suggesting that this is a simple root (multiplicity 1).
   - At \( x = -1 \), the graph touches the x-axis and turns around, indicating a double root (multiplicity 2).
   - At \( x = 1 \), the graph also touches the x-axis and turns around, indicating another double root (multiplicity 2).

2. **Polynomial degree**:
   - The graph shows a total of 5 roots considering multiplicities (1 from \( x = -4 \), 2 from \( x = -1 \), and 2 from \( x = 1 \)). Therefore, the degree of the polynomial is 5.

Now, let's check the options:
- The correct equation should reflect the behavior at each intercept, meaning:
  - \( (x + 4) \) as a linear factor (for \( x = -4 \)),
  - \( (x + 1)^2 \) as a quadratic factor (for \( x = -1 \)),
  - \( (x - 1)^2 \) as a quadratic factor (for \( x = 1 \)).

The correct equation should be:
\[
p(x) = (x + 4)(x + 1)^2(x - 1)^2
\]

This corresponds to **Option C**.

Let me know if you want further details or have any questions. Here are 5 related questions to explore further:

1. What is the general form of a polynomial of degree 5?
2. How can we determine the multiplicity of a root from the graph of a polynomial?
3. What effect does the multiplicity of a root have on the shape of a polynomial's graph?
4. How can we use the leading coefficient to determine the end behavior of a polynomial function?
5. How does factoring help in determining the roots of a polynomial?

**Tip**: Always check the behavior of the graph near the intercepts to understand the multiplicity of each root.

¿Cuál puede ser la ecuación de p(x) basada en la gráfica del polinomio?

Learn how to identify the equation of a polynomial p(x) from its graph by analyzing the roots, their multiplicities, and the behavior near the intercepts. This problem involves concepts such as polynomial degrees, roots, and graph analysis, suitable for students in Grades 10-12.

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