<s>[INST] Factorisation of $4x^4-4x^3+4x^2+2x+1$ with real coefficients 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 
 [/INST] 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 $(x - r)(x - \bar{r})$
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