1. **State the problem:** Solve the quadratic equation $4x^2 - 5x - 12 = 0$. 2. **Formula used:** The quadratic formula is given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a$, $b$, and $c$ are coefficients from the quadratic equation $ax^2 + bx + c = 0$. 3. **Identify coefficients:** Here, $a = 4$, $b = -5$, and $c = -12$. 4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-5)^2 - 4 \times 4 \times (-12) = 25 + 192 = 217$$ 5. **Apply the quadratic formula:** $$x = \frac{-(-5) \pm \sqrt{217}}{2 \times 4} = \frac{5 \pm \sqrt{217}}{8}$$ 6. **Final answer:** The solutions are $$x = \frac{5 + \sqrt{217}}{8} \quad \text{and} \quad x = \frac{5 - \sqrt{217}}{8}$$ These are the two roots of the quadratic equation.