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. 
 A: Let $f(x) = x^4 - x^3 +4x^2 +3x+5$. Once you know that $f(x)$ doesn't have real solutions, try some easy complex numbers like $i$, $\omega$, other $n^{th}$ roots of unity etc. Note that if we plug in $x = \omega$, where $\omega$ is the complex cube-root of unit, we get that $$f(\omega) = \omega^4 - \omega^3 +4 \omega^2 + 3 \omega + 5 = \omega - 1 + 4 \omega^2 + 3 \omega +5 = 4(\omega^2 + \omega + 1) = 0.$$
Hence, $x^2 + x + 1$ divides $f(x)$. Hence, $f(x) = (x^2 + x + 1) (x^2 + ax + b)$. This gives us that $$f(x) = x^4 + (a+1)x^3 + (1 + a + b)x^2 +(a+b)x + b$$
Hence, $a=-2$ and $b=5$. Hence, $$f(x) =  (x^2 + x + 1) (x^2 - 2x + 5)$$
Hence, the roots are $$x = \omega, \omega^2, 1 \pm 2i$$ 
A: Note that your polynomial is equal to $(x^2-2x+5)(x^2+x+1)$.
Remark: Either the polynomial factors nicely over the integers, or we have kind of a mess.  If it factors nicely, then the constant terms of the factors are $1$ and $5$, or $-1$ and $-5$.
Let's be positive and test  $1$ and $5$ first. Let the coefficients of $x$ in the factors be $a$ and $b$ respectively. The coefficient of $x^3$ in the product is then $a+b$. This must be $-1$. The coefficient of $x$ in the product is $5a+b$. This must be $3$. Solve. If we are lucky, the coefficient of $x^2$ in the product will turn out to be $4$. It is.   
A: $$x^4 - x^3 + 4x^2 + \underbrace{3x}_{4x-x} + \overbrace{5}^{4+1} = \\\color{red}{x^4-x^3}+4x^2+4x+4\color{red}{-x+1}\\={x^4-x^3}-x+1+4(x^2+x+1)\\={x^3(x-1)}-(x-1)+4(x^2+x+1)\\=(x-1)(x^3-1)+4(x^2+x+1)\\=(x-1)(x-1)(x^2+x+1)+4(x^2+x+1)\\=(x^2+x+1)(x-1)^2+4)=(x^2+x+1)(x^2-2x+5)$$
As for the roots, I assume you could solve those two quadratic equations and you could find the results on Wolfram|Alpha.
A: 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.
