1. **Problem Statement:** We need to find a polynomial function that matches the given graph. 2. **Observations:** The graph has roots approximately at $x = -6$, $x = 0$, and $x = 4$. The graph opens downward at both ends, indicating a negative leading coefficient and an even degree polynomial (likely degree 4). 3. **General Form:** A polynomial with roots at $x = -6$, $x = 0$, and $x = 4$ can be written as: $$y = a(x + 6)^m x^n (x - 4)^p$$ where $m, n, p$ are the multiplicities of the roots and $a$ is the leading coefficient. 4. **Degree and Multiplicities:** Since the graph looks like a 4th degree polynomial and the root at $x=4$ appears to have multiplicity 2 (the graph touches the x-axis and turns), we can set $m=1$, $n=1$, and $p=2$. 5. **Leading Coefficient:** The graph opens downward, so $a < 0$. We can choose $a = -1$ for simplicity. 6. **Final Function:** $$y = - (x + 6) x (x - 4)^2$$ This function has roots at $-6$, $0$, and $4$ (with multiplicity 2), degree 4, and opens downward, matching the graph. 7. **Verification:** Expanding the function: $$y = - (x + 6) x (x^2 - 8x + 16) = - x (x + 6)(x^2 - 8x + 16)$$ First expand $(x + 6)(x^2 - 8x + 16)$: $$x^3 - 8x^2 + 16x + 6x^2 - 48x + 96 = x^3 - 2x^2 - 32x + 96$$ Then multiply by $x$: $$x^4 - 2x^3 - 32x^2 + 96x$$ Apply the negative sign: $$y = -x^4 + 2x^3 + 32x^2 - 96x$$ This polynomial matches the described behavior. **Answer:** $$y = - (x + 6) x (x - 4)^2$$