The problem asks for a polynomial function $f(x)$ in factored form that matches the graph, with $a = 1$.

Looking at the given function: $f(x) = (x + 7)(x + 2)(x - 1)(x - 2)$ we see that it has roots at $x = -7$, $x = -2$, $x = 1$, and $x = 2$, as expected from each factor. These roots align with the points where the graph intersects the x-axis.

This polynomial likely matches the graph, given the shape and intercepts. Let me know if you need further details or explanations on the polynomial structure.

Would you like additional details or have any questions?

Here are some related questions to delve further into polynomial functions:

1. How do we determine the degree of a polynomial from its factored form?
2. What is the impact of each factor on the shape of the polynomial curve?
3. How does changing the leading coefficient $a$ affect the graph's orientation?
4. How can we verify that the polynomial accurately represents the graph?
5. What are the possible turning points for a polynomial of this degree?

Tip: The roots of a polynomial in factored form give the x-intercepts of its graph directly.