Answer
(a) $f(x)=(x^2+1+\sqrt 2 x)(x^2+1-\sqrt 2 x)$
(b) $\frac{-\sqrt 2\pm i\sqrt {2}}{2}, \frac{\sqrt 2\pm i\sqrt {2}}{2}$
Work Step by Step
(a) $f(x)=x^4+1=x^4+2x^2+1-2x^2=(x^2+1)^2-2x^2=(x^2+1+\sqrt 2 x)(x^2+1-\sqrt 2 x)$
(b) Solve $x^2+\sqrt 2 x+1=0$ to get $x=\frac{-\sqrt 2\pm\sqrt {2-4}}{2}=\frac{-\sqrt 2\pm i\sqrt {2}}{2}$
Solve $x^2-\sqrt 2 x+1=0$ to get $x=\frac{\sqrt 2\pm\sqrt {2-4}}{2}=\frac{\sqrt 2\pm i\sqrt {2}}{2}$
