The graph provided shows the polynomial $f(x)$, and we are told that $f(x)$ is a degree 4 polynomial. From the graph, we can extract the following key features:

1. Degree 4 Polynomial: This means the polynomial has 4 roots (including multiplicities).
2. Roots (x-intercepts): From the graph, the polynomial crosses the x-axis at $x = -2$, $x = 0$, and $x = 2$.
   • At $x = -2$, the graph appears to "bounce off" the x-axis, indicating this is a root with even multiplicity (likely multiplicity 2).
   • At $x = 0$ and $x = 2$, the graph crosses the x-axis, indicating these are roots with odd multiplicity (likely multiplicity 1).
3. End Behavior: Since the graph falls as $x \to \infty$ and rises as $x \to -\infty$, the leading coefficient of the polynomial must be negative.

Factored Form of $f(x)$:

Given these observations, we can write the factored form of $f(x)$ as: $f(x) = -a(x + 2)^2(x)(x - 2)$ where $a$ is a constant that can be determined by considering a specific point on the graph.

Determining $a$:

Looking at the graph, we notice that the polynomial passes through the point $(1, 2)$. We can substitute $x = 1$ and $f(1) = 2$ into the factored form to determine $a$.

$f(1) = -a(1 + 2)^2(1)(1 - 2)$ $2 = -a(3)^2(1)(-1)$ $2 = -a(9)(-1)$ $2 = 9a$ $a = \frac{2}{9}$

Thus, the factored equation for $f(x)$ is: $f(x) = -\frac{2}{9}(x + 2)^2x(x - 2)$

Would you like more details, or do you have any further questions?

Here are some related questions you could explore:

1. How do you verify the degree of a polynomial from its graph?
2. What is the significance of the multiplicity of roots in a polynomial?
3. How do you determine the leading coefficient of a polynomial from its graph?
4. How can we find more precise values for specific points on a polynomial graph?
5. What would change in the polynomial if the end behavior was reversed?

Tip: When analyzing a polynomial graph, always look at the behavior at the x-intercepts to determine root multiplicities.