Possible "clever" ways to solve $x^4+x^3-2x+1=0$, with methodological justification 
Solve the quartic polynomial :
$$x^4+x^3-2x+1=0$$
where $x\in\Bbb C$.
Algebraic, trigonometric and all possible methods are allowed.


I am aware that, there exist a general quartic formula. (Ferrari's formula). But, the author says, this equation doesn't require general formula. We need some substitutions here.
I realized there is no any rational root, by the rational root theorem.
The harder part is, WolframAlpha says the factorisation over $\Bbb Q$ is impossible.
Another solution method can be considered as the quasi-symmetric equations approach. (divide by $x^2$).
$$x^2+\frac 1{x^2}+x-\frac 2x=0$$
But the substitution $z=x+\frac 1x$ doesn't make any sense.
I want to ask the question here to find possible smarter ways to solve the quartic.

HINT.
$$4(x^4+x^3-2x+1)=(2x^2+x-1)^2+3(x-1)^2$$
$$=\big(2x^2+(1+i\sqrt3\,)(x-1)\big)\big(2x^2+(1-i\sqrt3)(x-1)\big),$$
DETAILS.

*

*The first quadratic term can be chosen in the form of $(2x^2+x+a)^2,$ which provides the coefficients near $x^4$ and $x^3.$


*The difference between the left part and the first quadratic term should be a full square.
