1. **State the problem:** We need to determine which graph could represent the function $$f(x) = (x + 2)(x - 6)(x^2 + 1)$$ based on the given descriptions. 2. **Analyze the function:** The function is a product of three factors: two linear factors $(x+2)$ and $(x-6)$, and one quadratic factor $(x^2 + 1)$. 3. **Identify roots:** The roots of $f(x)$ come from the linear factors since $x^2 + 1 = 0$ has no real roots. - Roots are at $x = -2$ and $x = 6$. 4. **Behavior at roots:** Since both linear factors are to the first power, the graph crosses the x-axis at $x = -2$ and $x = 6$. 5. **Degree and end behavior:** The degree of $f(x)$ is 4 (two linear factors and one quadratic factor). - Leading term is $x \cdot x \cdot x^2 = x^4$ with positive coefficient. - As $x \to \pm \infty$, $f(x) \to +\infty$. 6. **Check graph options:** - Graph A has x-intercepts approximately at $-2$ and $6$, crosses the x-axis twice, and has a local minimum between the roots. - Graph B has only one x-intercept at $6$, no root near $-2$. - Graph C and D show partial graphs with no clear x-intercepts visible. 7. **Conclusion:** Since $f(x)$ must have exactly two real roots at $-2$ and $6$, and the graph crosses the x-axis at these points, **Graph A** matches the function's behavior. **Final answer:** Graph A