How to solve this quartic equation? For the quartic equation:
$$x^4 - x^3 + 4x^2 + 3x + 5 = 0$$
I tried Ferrari so far and a few others but I just can't get its complex solutions. I know it has no real solutions. 

Use the magic quartic formula!
$$a = -1, b = 4, c = 3, d = 5$$
$$u = -\frac{29}{12}, \Delta_0 = 85, \Delta_1 = -826, Q = \sqrt[3]{666i-541}$$
The discriminant is 65712, meaning that there are either four real or four complex roots.
$$v = \frac{(\sqrt[3]{666i-541})^2 + 85}{3\sqrt[3]{666i-541}} = \frac{(666i-541)\sqrt[3]{666i-541} + 85(\sqrt[3]{666i-541})^2}{3(666i-541)}$$
$$=\frac{736237\sqrt[3]{666i-541} - 85(666i+541)(\sqrt[3]{666i-541})^2}{2208711}$$
And you should be able to take it from there.
