The graph you've provided is a polynomial function, and we need to determine its equation with the least degree that matches the key features of the graph. Let's break this down based on the characteristics observed:

1. Roots (x-intercepts): The graph crosses or touches the x-axis at the following points:

   • $x = -3$ (crosses, so it's a root of odd multiplicity, likely 1)
   • $x = -1$ (touches and turns, indicating a root of even multiplicity, likely 2)
   • $x = 2$ (crosses, so it's a root of odd multiplicity, likely 1)

2. End behavior:

   • As $x \to -\infty$, $f(x) \to \infty$ (upward).
   • As $x \to \infty$, $f(x) \to -\infty$ (downward).

   This indicates that the leading coefficient is negative, and the degree of the polynomial is odd.

3. Degree of the polynomial: Since there are three distinct roots and one of them has multiplicity 2, the degree of the polynomial should be at least 4. Specifically, the degree is likely 4 due to the structure of the graph (two turning points and the end behavior described).

Thus, a polynomial of least degree with these characteristics would be: $f(x) = -a(x + 3)(x + 1)^2(x - 2)$ where $a$ is a constant that determines the vertical stretch/compression.

We can conclude that the polynomial will have a negative leading coefficient based on the end behavior.

Do you have any specific questions or want a more detailed explanation of any part?

Here are 5 related questions for further exploration:

1. How do multiplicities of roots affect the shape of a graph near its x-intercepts?
2. What determines the end behavior of polynomial functions?
3. How can we determine the leading coefficient by just looking at a graph?
4. How do we confirm the degree of a polynomial given its graph?
5. What is the significance of turning points in determining the degree of a polynomial?

Tip: The number of turning points in the graph of a polynomial can help you estimate its degree. A degree $n$ polynomial can have up to $n-1$ turning points.