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The question asks which of the given formulas corresponds to the factored expression of the function shown in the graph. Let's analyze the graph and the provided answer options to find the correct one.

### Analyzing the Graph

1. **Roots (x-intercepts):**  
   The graph intersects the x-axis at \(x = -3\), \(x = 2\), and \(x = 5\).  
   These points indicate that the function has factors of \((x + 3)\), \((x - 2)\), and \((x - 5)\).

2. **Shape of the Graph (Behavior at Roots):**  
   - The graph touches the x-axis at \(x = 2\) and does not cross it, indicating a **double root** at \(x = 2\).  
   - The graph crosses the x-axis at \(x = -3\) and \(x = 5\), indicating these are **single roots**.

3. **Leading Coefficient:**  
   The function has a positive leading coefficient because it opens upwards at the end.

### Determining the Correct Option

Given the roots and their multiplicities:
- The factor for \(x = -3\) is \((x + 3)\).
- The factor for \(x = 2\) is \((x - 2)^2\) because it is a double root.
- The factor for \(x = 5\) is \((x - 5)\).

Thus, the function must be of the form:  
\[
f(x) = k \cdot (x + 3) \cdot (x - 2)^2 \cdot (x - 5)
\]  
where \(k\) is a constant.

#### Checking the Answer Choices:

a) \(m(x) = 300 \cdot (x + 3) \cdot (x - 2) \cdot (x - 5)^2\)  
   - Incorrect: The root at \(x = 5\) should not be a double root; it should be a single root.

b) \(n(x) = (x + 3) \cdot (x - 2) \cdot (x - 5)\)  
   - Incorrect: The root at \(x = 2\) is a double root, not a single root.

c) \(q(x) = -2 \cdot (x + 3) \cdot (x - 2) \cdot (x - 5)^2\)  
   - Incorrect: The coefficient is negative, and \(x = 5\) is incorrectly shown as a double root.

d) \(r(x) = 2 \cdot (x + 3) \cdot (x - 2)^2 \cdot (x - 5)\)  
   - **Correct**: This matches the factors and the multiplicities of the roots correctly.

e) \(g(x) = -2 \cdot (x + 3) \cdot (x - 2) \cdot (x - 5)^2 \cdot 300\)  
   - Incorrect: The expression has the wrong multiplicities, a negative leading coefficient, and an extra constant factor.

### Conclusion:

The correct answer is **d) \(r(x) = 2 \cdot (x + 3) \cdot (x - 2)^2 \cdot (x - 5)\)**.

Would you like more details on any step or have any questions? Here are some related topics that could help:

1. What is a polynomial function and its properties?
2. How to identify roots and their multiplicities from a graph?
3. Why does the leading coefficient affect the direction of a polynomial?
4. How to factor polynomial expressions?
5. Understanding double roots and their graphical representation.

**Tip:** When analyzing polynomial functions, always check for the multiplicity of roots and the sign of the leading coefficient to determine the correct expression.

¿Cuál de las siguientes fórmulas corresponde a la expresión factorizada de la función mostrada en el gráfico?

Analyze a polynomial function graph with roots at x = -3, x = 2 (double root), and x = 5. Learn how to determine the correct factored form based on the graph's x-intercepts and multiplicities.

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