This is a bizarre question I encountered on a homework sheet:

The only way I can see of doing this is to use complex numbers. I have this idea of writing a root as $\displaystyle "x^2-a"$ where a is a positive real to bypass this. However, I have no way of working out the roots!Every irreducible polynomial in $\displaystyle \Re$[x] is of degree 1 or 2. Express $\displaystyle x^4+1$ as a product of irreducible polynomials in$\displaystyle \Re$[x]