The graph provided in the image corresponds to the behavior of a polynomial function. To match the given equation to the graph, we analyze the following features:

1. Roots (x-intercepts):

   • The graph intersects the x-axis at $x = -1$ and $x = 1$.
   • The intercept at $x = -1$ shows a cubic behavior (flattened curve), indicating a multiplicity of 3.
   • The intercept at $x = 1$ shows a linear behavior, indicating a multiplicity of 1.

2. Leading term behavior:

   • The graph falls as $x \to -\infty$ and rises as $x \to \infty$, indicating a positive leading coefficient.

3. Possible equation structure: Using the information above, the general structure of the polynomial is: $f(x) = k(x+1)^3(x-1)$ where $k > 0$.

Analyzing the Options:

• $f(x) = x^2(x+1)^3(x-1)$: This is incorrect because the extra $x^2$ introduces a root at $x = 0$, which is not present in the graph.
• $f(x) = x(x+1)^3(x-1)$: This is incorrect because it introduces a root at $x = 0$, which is not on the graph.
• $f(x) = x(x-1)^3(x+1)$: Incorrect multiplicity and root order.
• $f(x) = (x+1)^3(x-1)$: Correct structure matching the graph.

Conclusion:

The correct equation is: $f(x) = (x+1)^3(x-1)$

Would you like further clarification or detailed reasoning? Here are five related questions you might find helpful:

1. How do we determine the multiplicity of a root from a graph?
2. Why does the sign of the leading coefficient determine the end behavior of the polynomial?
3. How can you verify polynomial roots algebraically?
4. What happens to the graph when the multiplicity of a root changes (e.g., from 1 to 2)?
5. How can we expand and simplify the equation to determine the degree of the polynomial?

Tip: Always double-check the behavior at each root to confirm the multiplicity matches the graph!