From the graph shown in the image, we can see that it is a cubic function (polynomial of degree 3) with its roots visible as follows:

1. The function crosses the x-axis at x = -2, x = 1, and x = 3.

Thus, the factored form of the polynomial must be based on the roots, so the equation is structured as: $f(x) = (x + 2)(x - 1)(x - 3)$

Looking at the provided options, the correct choice is: $\mathbf{f(x) = (x + 2)(x - 1)(x - 3)}$

This matches the behavior of the graph as it crosses the x-axis at the specified roots and has the correct shape for a cubic polynomial.

Would you like a detailed breakdown of why these roots and this factorization match the graph?

Here are five related questions to expand on this topic:

1. How do we determine the degree of a polynomial based on its graph?
2. What is the significance of the roots in a polynomial function's graph?
3. How can we verify if a polynomial equation is correct for a given graph?
4. How do end behaviors of polynomials differ based on their degrees?
5. Can a cubic polynomial have repeated roots, and how would this appear on a graph?

Tip: When identifying polynomial equations from graphs, always check the number of roots and the direction of the ends of the graph to confirm the degree and leading coefficient