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The problem you're working on asks you to identify the equation of the polynomial \( p(x) \) based on its graph. The choices are given in the first image, and the second image shows the graph of the polynomial.

To solve this, we need to match the graph with the factored form of the polynomial, which gives us information about the zeros (x-intercepts) and the behavior at those intercepts.

Let's break down the key features from the graph:
1. **The polynomial has three zeros** (the points where it crosses the x-axis). These appear to be at \( x = -5, -2, \) and \( x = 2 \).
2. The polynomial has a local minimum between \( -2 \) and \( -5 \), and another zero at \( x = 2 \), where it appears to touch the x-axis and then turn around.

### Analyzing the choices:
- In factored form, the zeros of a polynomial \( p(x) \) can be written as factors of the form \( (x - a) \), where \( a \) is the x-coordinate of the zero.
- From the graph, the zeros are at \( x = -5 \), \( x = -2 \), and \( x = 2 \).

Let's check the factored form of each equation:
- **Choice A:** \( p(x) = (x + 5)(2x + 3)(x - 2) \)  
  This has zeros at \( x = -5 \) and \( x = 2 \), but it doesn't seem to have a zero at \( x = -2 \).
  
- **Choice B:** \( p(x) = (x - 5)(2x - 3)(x + 2) \)  
  This has zeros at \( x = 5 \) and \( x = -2 \), but it doesn't match the graph since it doesn't have a zero at \( x = -5 \) and \( x = 2 \).

- **Choice C:** \( p(x) = (-5x)(\frac{3}{2} - x)(2x) \)  
  This choice has zeros at \( x = 0, \frac{3}{2}, \) and \( 2 \), which doesn't match the graph at all.

- **Choice D:** \( p(x) = (5x)(\frac{3}{2} x)(-2x) \)  
  This has zeros at \( x = 0 \) and other factors which do not match the zeros seen on the graph.

### Conclusion:
None of the provided choices fully match the graph's zeros. The graph clearly shows zeros at \( x = -5 \), \( x = -2 \), and \( x = 2 \), which suggests a possible misalignment or a need for revising the choices. Would you like to adjust your options based on these observations or need additional help with this?

What could be the equation of the polynomial p(x) based on its graph?

Learn how to determine the equation of a polynomial in factored form based on its graph. This problem covers algebraic concepts, including identifying zeros, analyzing factored forms, and applying the Zero Product Property. Suitable for high school students in Grades 10-11.

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