Here is a solution to the general quartic compactified. Given,
$$Ax^4+Bx^3+Cx^2+Dx+E = 0$$
divide by $A$ to get the simpler,
$$x^4+ax^3+bx^2+cx+d=0$$
Then the four roots are,
$$x_{1,2} = -\frac{a}{4}+\frac{\color{red}\pm\sqrt{u}}{2}\color{blue}+\frac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\color{red}\pm\sqrt{u}}}\tag1$$
$$x_{3,4} = -\frac{a}{4}+\frac{\color{red}\pm\sqrt{u}}{2}\color{blue}-\frac{1}{4}\sqrt{3a^2-8b-4u+\frac{-a^3+4ab-8c}{\color{red}\pm\sqrt{u}}}\tag2$$
where,
$$u = \frac{a^2}{4}-\frac{2b}{3} +\frac{1}{3}\left(v_1^{1/3}\zeta_3+\frac{b^2 - 3 a c + 12 d}{v_1^{1/3}\zeta_3}\right)$$
with $v_1$ any non-zero root of the quadratic,
$$v^2 + (-2 b^3 + 9 a b c - 27 c^2 - 27 a^2 d + 72 b d)v + (b^2 - 3 a c + 12 d)^3 = 0$$
and a chosen cube root of unity $\zeta_3^3 = 1$ such that $u$ is also non-zero. (Normally, just use $\zeta_3=1$, but not when $a^3-4ab+8c = 0$.)
P.S. This is essentially the method used by Mathematica, though much simplified for aesthetics.
It's much easier to work from the roots to the coefficients than the other way around.
If you have a cubic polynomial $ax^3+bx^2+cx+d$ with roots $\alpha$, $\beta$, $\gamma$, then consider
$$ a(x-\alpha)(x-\beta)(x-\gamma) $$
If you multiply this out you get a cubic polynomial that also has $\alpha$, $\beta$, $\gamma$ as roots and has $a$ as the leading coefficient.
And this polynomial has to be the same polynomial as the one you started with, because otherwise subtracting them would produce a nonzero polynomial or degree $2$ or less that has $3$ roots, which is impossible.
Thus, multiplying out the above formula will give you all the coefficients as functions of the roots. Dividing through with $a$ then produces, among other things, the formula you're after. And this works for any degree.
(If there are multiple roots, then the above argument doesn't work 100% -- but in order to fix it you'd need to have a better definition for what it means in the first place for the roots to be, say, 2,2,3 instead of 2,3,3).
Best Answer
As Moron says, Cardano works. You merely need to be a bit more careful than usual.