1. **Problem Statement:** We need to determine which polynomial matches the graph described. The graph has: - A repeated root at $-1$ with a local maximum (indicating even multiplicity and negative leading coefficient). - A root at $3$ where the graph crosses the x-axis (indicating odd multiplicity). 2. **Recall:** - A repeated root with even multiplicity causes the graph to touch and turn around at that root (a "flattened touch"). - A root with odd multiplicity causes the graph to cross the x-axis. - The sign of the leading coefficient affects the end behavior of the graph. 3. **Analyze each option:** - **A:** $f(x) = -(x + 1)^2(x - 3)$ - Root at $-1$ with multiplicity 2 (even), and negative leading coefficient. - Root at $3$ with multiplicity 1 (odd). - Matches the description: repeated root at $-1$ with local max (due to negative sign), crosses at $3$. - **B:** $f(x) = (x - 3)^2(x + 1)$ - Root at $3$ with multiplicity 2 (even), root at $-1$ with multiplicity 1 (odd). - Does not match the graph description. - **C:** $f(x) = (x - 1)(x + 3)^2$ - Roots at $1$ and $-3$, not matching $-1$ and $3$. - **D:** $f(x) = - (x + 3)(x - 3)^2$ - Roots at $-3$ and $3$, not matching $-1$ and $3$. 4. **Conclusion:** Option A matches the graph's roots, multiplicities, and behavior. **Final answer:** $$f(x) = -(x + 1)^2(x - 3)$$