The graph provided appears to be a cubic function with three real roots based on the way it intersects the x-axis. Let’s analyze the characteristics of this function to match it with the correct algebraic expression from the multiple-choice options.

Step-by-Step Analysis:

1. Roots of the Function:

   • The function intersects the x-axis at $x = -5$, which means $x + 5$ is a factor.
   • There are two other points of interest on the x-axis, suggesting that the remaining factor should include two real roots or complex roots. Observing the shape of the curve, it does not cross the x-axis at these two points, which implies a quadratic term without real roots.

2. Quadratic Term:

   • We need a quadratic term that has no real roots. The term $(x^2 + 4)$ fits this criteria, as $x^2 + 4 = 0$ has complex roots $x = \pm 2i$.
   • The term $(x^2 - 4)$ has real roots $x = \pm 2$, which would lead to additional x-intercepts on the graph, which we don’t see. Therefore, we can rule out options with $(x^2 - 4)$ in them.

3. Matching the Options:

   • Based on the analysis, the function should have factors that include $(x + 5)$ and $(x^2 + 4)$.
   • The correct answer is: $f(x) = (x + 5)(x^2 + 4)$

Answer

The correct option is: $\boxed{f(x) = (x + 5)(x^2 + 4)}$

Would you like a more detailed breakdown or have further questions?

Here are some related questions to expand on this topic:

1. How can you determine the degree of a polynomial function based on its graph?
2. What are the visual differences on a graph between a quadratic factor with real roots and one with complex roots?
3. How does the behavior near the roots reflect the multiplicity of each root in a polynomial function?
4. Why does a polynomial with complex roots not intersect the x-axis at those roots?
5. What are the characteristics of even-degree and odd-degree polynomial functions based on their end behaviors?

Tip: When analyzing a polynomial function's graph, look for where the function intersects the x-axis to identify real roots, and observe the end behavior to determine the degree of the polynomial.