How could I know that $X^4+1$ is $(X^2+\sqrt 2X+1)(X^2-\sqrt 2X+1)$? I thought that $X^4+1$ was irreducible, but in fact, $$X^4+1=(X^2+\sqrt 2X+1)(X^2-\sqrt 2X+1).$$
In general, how can I have the intuition of such a factorisation if I don't know it ?

We know that $$a^4+4=a^4+4a^2+4-4a^2=(a^2+2)^2-(2a)^2=(a^2+2a+2)(a^2-2a+2)$$
This is an well known identity, most easily identifiable from the difference between two squares. It is called the Sophie Germain Identity= 
Putting in $x=\sqrt{2} a$, we have that $$x^2+1=(x^2+\sqrt{2} x+1)(x^2-\sqrt{2}x+1)$$
Though the first step is unnecessary, I added it as it is a generally useful formula. 
