1. **State the problem:** We need to determine which polynomial among the given options matches the graph of $P(x)$ based on its roots and their multiplicities. 2. **Analyze the graph:** The graph crosses the x-axis at $x = -1$ and $x = 2$, and also touches or crosses at $x = 0$. The behavior near these roots suggests: - At $x = -1$, the graph touches the x-axis and turns around, indicating a root with even multiplicity (likely 2). - At $x = 2$, the graph also touches and turns, indicating another root with even multiplicity (likely 2). - At $x = 0$, the graph crosses the x-axis, indicating a root with odd multiplicity (likely 1). 3. **Recall polynomial root multiplicity rules:** - If the graph crosses the x-axis at a root, the root has odd multiplicity. - If the graph touches and turns at the x-axis, the root has even multiplicity. 4. **Check each polynomial option:** - $p(x) = x(x + 1)^2(x - 2)^2$: roots at $0$ (multiplicity 1), $-1$ (multiplicity 2), $2$ (multiplicity 2). Matches the graph behavior. - $p(x) = x(x + 1)^2(x + 2)^2$: roots at $0$, $-1$, and $-2$. The graph does not show a root at $-2$. - $p(x) = x(x - 1)^2(x - 2)^2$: roots at $0$, $1$, and $2$. The graph does not show a root at $1$. - $p(x) = (x + 1)^2(x - 2)^2$: roots at $-1$ and $2$ only, missing root at $0$. 5. **Conclusion:** The polynomial $p(x) = x(x + 1)^2(x - 2)^2$ best matches the graph. **Final answer:** $$p(x) = x(x + 1)^2(x - 2)^2$$