Solve in exact form: $x^6 - x^5 + 4 x^4 - 4 x^3 + 4 x^2 - x + 1=0$ ( WolframAlpha failed) 
Solve the polynomial in closed form:
$$x^6 - x^5 + 4 x^4 - 4 x^3 + 4 x^2 - x + 1=0$$

WolframAlpha obviously failed.
I tried several ways:

*

*I tried the Rational Root Thereom, but there is no rational root.

*I tried possible factorisations

$$x^6-x^5+4x^4-4x^3+4x^2-x+1= (x^2+a_1x+a_2)(x^4+a_3x^3+a_4x^2+a_5x+a_6)$$
and
$$x^6-x^5+4x^4-4x^3+4x^2-x+1= (x^3+a_1x^2+a_2x+a_3)(x^3+a_4x+a_5)$$
But, the expansions are terrible!

The problem can be rewritten as $$x((x-1)x+1)(x^3+3x-1)+1=0$$ Now solving this we get $6$ roots:
$1.$ $-0.1399-1.7659i$
$2.$ $-0.1399+1.7659i$
$3.$ $-0.04270-0.56305i$
$4.$ $-0.04270+0.56305i$
$5.$ $0.67660-0.73635i$
$6.$ $0.67660+0.73635i$
These are all approximations I got from a software. Strict factorisation $($writing expression only in terms of factors$)$ is not really possible in my opinion or you can use these approx roots for that.
