1. **State the problem:** Solve the quadratic equation $$-4x^2 - x + 4 = 0$$ and find the greatest solution. 2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0$$, the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a = -4$, $b = -1$, and $c = 4$. 3. **Calculate the discriminant:** $$b^2 - 4ac = (-1)^2 - 4(-4)(4) = 1 + 64 = 65$$ 4. **Apply the quadratic formula:** $$x = \frac{-(-1) \pm \sqrt{65}}{2(-4)} = \frac{1 \pm \sqrt{65}}{-8}$$ 5. **Simplify the expression:** $$x = \frac{1 \pm \sqrt{65}}{-8} = -\frac{1 \pm \sqrt{65}}{8}$$ 6. **Write the two solutions explicitly:** $$x_1 = -\frac{1}{8} + \frac{\sqrt{65}}{8}$$ $$x_2 = -\frac{1}{8} - \frac{\sqrt{65}}{8}$$ 7. **Identify the greatest solution:** Since $\sqrt{65} > 0$, the greatest solution is $$x_1 = -\frac{1}{8} + \frac{\sqrt{65}}{8}$$ **Final answer:** The greatest solution is option c. $-\frac{1}{8} + \frac{\sqrt{65}}{8}$.