[Solved] Multiplying imaginary factors of a polynomial

The problem:

$\displaystyle (x - e^{\frac{i\pi}{4}})(x - e^{\frac{-i\pi}{4}})$

My attempt:

I used the distributive rule to expand it, and got:

$\displaystyle x^2 - x(e^{\frac{i\pi}{4}} + e^{\frac{-i\pi}{4}}) + 1$

I recognise that $\displaystyle cos(\frac{\pi}{4}) = \frac{1}{2}(e^{\frac{i\pi}{4}} + e^{\frac{-i\pi}{4}})$

So I get $\displaystyle x^2 - x(2cos(\frac{\pi}{4})) + 1$

=$\displaystyle x^2 - \frac{2x}{\sqrt{2}} + 1$

=$\displaystyle x^2 - \sqrt{2}x + 1$

I'm unsure if this is correct, since I don't have a solution. Any assistance would be greatly appreciated.