Can this equation be solved by radicals? $x^6+x^5+2x^4-2x^3-x^2+1=0$ Can this equation be solved by radicals? If it can be solved, what is the method? At least, I will resolve it myself if you specify reference or source.

$$x^6+x^5+2x^4-2x^3-x^2+1=0$$


Yes because$$x^6+x^5+2x^4-2x^3-x^2+1=0\iff (x^3-1)^2+x^2(x^3-1)+2x^4=0$$ It follows solving as a quadratic in $x^3-1$
$$2(x^3-1)=-x^2\pm\sqrt{-7}x\iff2x^3+x^2\mp\sqrt{-7}x^2-2=0$$ so you can apply Cardano's formula.
