I have used many mathematical trick to factorise this polynomial : $4x^4-4x^3+4x^2+2x+1$ with real coeffecients but i didn't succeed because as I see all it's root are complex , and I want if there is any suitable method to factorise it ? and thanks in advanced .
Note: The motivation of this question is to compute some integral
Let $$ x = \frac{i t }{ \sqrt 2} $$
Your polynomial is now $$ t^4 + i \sqrt 2 t^3 - 2 t^2 + i \sqrt 2 t + 1 $$ Divide by $t^2$ and then introduce $w = t + \frac{1}{t}$
dividing by $t^2$ $$ t^2 + i \sqrt 2 t - 2 + \frac{i \sqrt 2}{t} + \frac{1}{t^2} $$ which is $$ w^2 + i \sqrt 2 w - 4 $$
so the roots are $$ w = \frac{- i \sqrt 2 \pm \sqrt{14}}{2} $$ We get back to four roots in $t$ with two quadratics, $$ t^2 + \frac{ i \sqrt 2 \pm \sqrt{14}}{2} t + 1 = 0 $$
Finally we get four $x$ roots in conjugate pairs giving real quadratics as $(x - r)(x - \bar{r})$