1. **State the problem:** We have a polynomial $P(x)$ with the following properties: - 4th finite differences are constant and equal to 48. - Exactly three distinct x-intercepts at $x=-3$, $x=-1$, and $x=2$. - The intercept at $x=-3$ is a local minimum with multiplicity 2. - The y-intercept is at $(0,-36)$. - The leading coefficient is 2. 2. **Determine the degree:** Since the 4th finite differences are constant and nonzero, $P(x)$ is a polynomial of degree 4. 3. **Form the polynomial:** Given the roots and multiplicities: - Root at $x=-3$ with multiplicity 2 (because it's a local minimum and intercept). - Roots at $x=-1$ and $x=2$ with multiplicity 1 each. So, $$P(x) = a(x+3)^2(x+1)(x-2)$$ where $a$ is the leading coefficient. 4. **Use the leading coefficient:** Given $a=2$, so $$P(x) = 2(x+3)^2(x+1)(x-2)$$ 5. **Check the y-intercept:** Evaluate $P(0)$: $$P(0) = 2(0+3)^2(0+1)(0-2) = 2 \times 9 \times 1 \times (-2) = -36$$ This matches the given y-intercept. 6. **Check behavior as $x \to \infty$:** Since the leading term is $2x^4$, as $x \to \infty$, $P(x) \to \infty$. 7. **Check the number of turning points:** A degree 4 polynomial can have up to 3 turning points. Here, the local minimum at $x=-3$ and the shape suggest exactly 3 turning points. 8. **Check 3rd finite differences:** The 4th finite differences are constant and equal to 48, so the 3rd finite differences are not constant but increasing. 9. **Check symmetry:** The polynomial is not symmetric about the origin (not odd function), so no point symmetry. **Final conclusions:** - Degree is 4. - Absolute minimum exists (local minimum at $x=-3$). - 3rd finite differences are not constant. - Exactly three turning points. - Multiplicity 2 at $x=-3$. - $P(x) \to \infty$ as $x \to \infty$. - Y-intercept at $(0,-36)$. - Range is not all real numbers. - No point symmetry about origin. - Leading coefficient is 2. **Answer:** All checked statements are correct as per the problem.