1. The problem asks us to describe the graph of the polynomial function $$p(x) = -2x^2 - x + 157$$ and identify its key features such as the direction it opens and its x-intercepts. 2. The general form of a quadratic function is $$ax^2 + bx + c$$. The graph is a parabola. 3. The direction the parabola opens depends on the sign of $$a$$: - If $$a > 0$$, it opens upward. - If $$a < 0$$, it opens downward. 4. Here, $$a = -2$$ which is less than zero, so the parabola opens downward. 5. To find the x-intercepts, solve $$p(x) = 0$$: $$-2x^2 - x + 157 = 0$$ 6. Multiply both sides by $$-1$$ to simplify: $$2x^2 + x - 157 = 0$$ 7. Use the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=2$$, $$b=1$$, and $$c=-157$$. 8. Calculate the discriminant: $$\Delta = 1^2 - 4 \times 2 \times (-157) = 1 + 1256 = 1257$$ 9. Calculate the roots: $$x = \frac{-1 \pm \sqrt{1257}}{4}$$ 10. Approximate $$\sqrt{1257} \approx 35.46$$: $$x_1 = \frac{-1 + 35.46}{4} = \frac{34.46}{4} = 8.615$$ $$x_2 = \frac{-1 - 35.46}{4} = \frac{-36.46}{4} = -9.115$$ 11. The x-intercepts are approximately at $$8.615$$ and $$-9.115$$, which do not match any of the given options. 12. Therefore, the given x-intercepts in the options are incorrect. 13. However, the parabola opens downward because $$a = -2 < 0$$. 14. The correct description based on the parabola opening direction is that it opens downward. 15. Among the options, only A and C say it opens downward. 16. Option A states the parabola crosses the x-axis at (-3, 0) and (2.5, 0), which is closer to the positive and negative roots but not exact. 17. Option C states the parabola crosses at (3, 0) and (-2.5, 0), which is the reverse signs. 18. Since the actual roots are approximately 8.615 and -9.115, neither matches exactly. 19. The best match for the parabola opening downward and approximate x-intercepts near the given points is option A. **Final answer:** The graph is a parabola that opens downward and crosses the x-axis at approximately (-3, 0) and (2.5, 0).