Factoring $x^{4} +1$, using real factoring to the second degree Factoring to the second degree using real numbers
$$x^{4} +1$$
I know that $ x^{4} +1=(x^{2} + i)(x^{2}-i).\;$ But these are complex but I thought using these in some kind of way? Got no where! 
And then I tried to guess, two solutions are $\pm (-1)^{1/4},\, $
which gave me $(x-\sqrt{i})(x+\sqrt{i}).\;$ But I have a feeling that I am really off here...
Hmm... Thinking! 
What kind of techniques do you use?  

Hint: Write $x^4 + 1 = (x^2 +1)^2 - (\sqrt{2}x)^2$
