First off I tried formatting this as best as I could (still new to LaTeX). My lecturer showed an example of a technique to factor polynomials with a degree $\ge$ 4 but I am having trouble understanding it.
We have that $ \deg p \geq 4, \deg p = r + s $ in the following
$$ p(x) = (a_r x ^r)+...+a_0)(b_sx^s+...+b_0) $$
We want to examine:
$$\begin{aligned} p(x) &= x^4 + 1 \\ &= (x^2 + ax + b) (x^2 + cx + d) \\ &=x^4+(a+c) x^3 + (ac+d+b)x^2 + (ad+bc)x + bd \end{aligned}$$
Coeffiant comparison:
I: a + c = 0
II: ac + b + d = o
III: ad + bc = 0
IV: bd = 1
$ I\Rightarrow \;a=-c \; ^{III}_{\Rightarrow}\; b=d \;(if \; c\ne0)\; ^{IV}_{\Rightarrow}\; d^2=1 \Rightarrow d=\pm 1$ Choose d=1
$II \Rightarrow c^2 = 2 \Rightarrow c= \pm\sqrt{2}$
Choose c= $\sqrt{2} \Rightarrow a=-\sqrt{2} $
$ \Rightarrow x^4+1 = (x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$