1. **State the problem:** Solve the quadratic equation $$4x^2 + 5x - 8 = 0$$. 2. **Formula used:** The quadratic formula for solving $$ax^2 + bx + c = 0$$ is $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$. 3. **Identify coefficients:** Here, $$a = 4$$, $$b = 5$$, and $$c = -8$$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = 5^2 - 4 \times 4 \times (-8) = 25 + 128 = 153$$. 5. **Apply the quadratic formula:** $$x = \frac{-5 \pm \sqrt{153}}{2 \times 4} = \frac{-5 \pm \sqrt{153}}{8}$$. 6. **Simplify the expression:** Since $$\sqrt{153}$$ cannot be simplified further, the solutions are $$x_1 = \frac{-5 + \sqrt{153}}{8}$$ and $$x_2 = \frac{-5 - \sqrt{153}}{8}$$. 7. **Final answer:** The solution set is $$\left\{ \frac{-5 + \sqrt{153}}{8}, \frac{-5 - \sqrt{153}}{8} \right\}$$.