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To find the correct equation for the polynomial \( p \) shown in the graph, let's analyze the key features of the graph:

### Key Points from the Graph:
1. **Zeros of the polynomial:** The graph intersects the x-axis at \( x = -1 \), \( x = -2.5 \), and \( x = 3 \).
2. **Behavior at the zeros:**
   - At \( x = -1 \), the graph touches the x-axis, indicating a **double root**.
   - At \( x = -2.5 \), the graph crosses the x-axis, indicating a **single root**.
   - At \( x = 3 \), the graph touches the x-axis, indicating a **double root**.

Based on these observations, the factors of the polynomial should reflect the nature of each root:
- A double root at \( x = -1 \) is represented by \( (x + 1)^2 \).
- A single root at \( x = -2.5 \) corresponds to \( (2x + 5) \) (note the transformation since \(-2.5\) can be expressed as \(-\frac{5}{2}\)).
- A double root at \( x = 3 \) is represented by \( (x - 3)^2 \).

### Matching with the Given Options:
- **Option A:** \( p(x) = (x + 1)^2(2x + 5)(x - 3) \) — Does not match as it does not have a double root at \( x = 3 \).
- **Option B:** \( p(x) = (x + 1)(2x + 5)(x - 3)^2 \) — Does not match as it does not have a double root at \( x = -1 \).
- **Option C:** \( p(x) = (x + 1)(2x + 5)(x - 3) \) — Does not match as it misses both double roots.
- **Option D:** \( p(x) = (x + 1)^2(2x + 5)(x - 3)^2 \) — Matches exactly with double roots at \( x = -1 \) and \( x = 3 \), and a single root at \( x = -2.5 \).

Thus, the correct answer is **Option D**.

Would you like any further details or explanations?

### Related Questions:
1. How do you determine the multiplicity of roots from a graph?
2. What is the impact of root multiplicity on the shape of a polynomial graph?
3. How can you factorize a polynomial with given roots and multiplicities?
4. What is the general form of a polynomial given its roots?
5. How do transformations affect the factors of a polynomial equation?

**Tip:** For a clearer understanding of root multiplicity, observe how the graph behaves near each root: crossing implies odd multiplicity, while touching without crossing implies even multiplicity.

¿Cuál puede ser la ecuación de p basada en el gráfico dado?

Analyze the graph of a polynomial function to determine the equation based on the roots and their multiplicities. The correct equation includes double roots at x = -1 and x = 3, and a single root at x = -2.5. Suitable for high school students.

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