1. **State the problem:** Find all complex solutions of the quadratic equation $$3x^2 - x + 4 = 0.$$\n\n2. **Recall the quadratic formula:** For an equation $$ax^2 + bx + c = 0,$$ the solutions are given by $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$\n\n3. **Identify coefficients:** Here, $$a = 3,$$ $$b = -1,$$ and $$c = 4.$$\n\n4. **Calculate the discriminant:** $$\Delta = b^2 - 4ac = (-1)^2 - 4 \times 3 \times 4 = 1 - 48 = -47.$$\n\n5. Since $$\Delta < 0,$$ the solutions are complex numbers.\n\n6. **Apply the quadratic formula:**\n$$x = \frac{-(-1) \pm \sqrt{-47}}{2 \times 3} = \frac{1 \pm \sqrt{-47}}{6}.$$\n\n7. **Simplify the square root of negative number:**\n$$\sqrt{-47} = i\sqrt{47},$$ where $$i$$ is the imaginary unit.\n\n8. **Write the final solutions:**\n$$x = \frac{1 \pm i\sqrt{47}}{6}.$$\n\n**Answer:** $$x = \frac{1 + i\sqrt{47}}{6}, \frac{1 - i\sqrt{47}}{6}.$$