1. The problem asks us to identify the equation of a polynomial $p(x)$ based on the shape and roots of its graph. 2. The polynomial's roots are at $x = 0$, $x = 3$, and $x = -\frac{7}{2}$ from the factors given: $x$, $(x-3)$, and $(2x+7)$. 3. Observing the graph's behavior at the roots helps us determine the powers of each factor: - At $x=0$, the graph crosses the x-axis, indicating an odd multiplicity (likely power 1). - At $x=3$, the graph shows a turning point, suggesting a repeated root (power 2). - At $x=-\frac{7}{2}$, the graph touches or crosses the axis, combined with its behavior indicates power 1 or 2. 4. The graph shows a flattened point near $x=3$, a key feature of a squared factor. 5. Among the given choices: - Option A: $p(x) = x(x-3)(2x+7)^2$ (single root at 3, squared at -7/2) - Option B: $p(x) = x^2(x-3)(2x+7)$ (double root at 0, single root at 3 and -7/2) - Option C: $p(x) = x(x-3)^2(2x+7)^2$ (squared roots at both 3 and -7/2) - Option D: $p(x) = x(x-3)^2(2x+7)$ (squared root at 3, single roots at 0 and -7/2) 6. The graph indicates a squared factor at $x=3$ and single at $x=0$ and $x=-\frac{7}{2}$. 7. Therefore, option D, $p(x) = x(x-3)^2(2x+7)$, matches the graph's shape and roots behavior. Final answer: $$p(x) = x(x - 3)^2(2x + 7)$$