The biquadratic equation, two of whose roots are $1+i, 1-\sqrt{2}$, is
The biquadratic equation, two of whose roots are $1+i, 1-\sqrt{2}$, is
$x^4-4 x^3+5 x^2-2 x-2=0$
$x^4+4 x^3-5 x^2+2 x+2=0$
$x^4+4 x^3-5 x^2+2 x-2=0$
$x^4+4 x^3+5 x^2-2 x+2=0$
Solution
When $1+i, 1-i$ are the roots, then
Sum of the roots $=1+i+1-i=2$
and product of the roots $=1+1=2$
The equation is $x^2-2 x+2=0$
When $1-\sqrt{2}, 1+\sqrt{2}$ are the roots, then
Sum of the roots $=1-\sqrt{2}+1+\sqrt{2}=2$
Product of the roots $=1-2=-1$
The equation is $x^2-2 x-1=0$
The bi-quadratic equation is
$\begin{array}{r}\left(x^2-2 x+2\right)\left(x^2-2 x-1\right)=0 \\ \left(x^4-(2+2) x^3=(-1+4+2) x^2\right. \\ =(2-4) x-2=0 \\ \Rightarrow \quad x^4-4 x^3+5 x^2-2 x-2=0\end{array}$