I was given the question $x^8 + 16$ and told to reduce it as much as able over the real numbers.
Here is what I tried
$x^8 + 16$
$(x^4+4)^2-8x^4$
$(x^4+4-2^{3/2}x^2)(x^4+4+2^{3/2}x^2)$
I can not figure where to go from here, I have also tried using complex conjugates and multiplying them to get it factored, which leads me to getting
$x^2-2^{3/2}x\cos{(\pi/8)}+2$ as a factor, which when I try to put in a calculator to expand further yields me the wrong result.
Can anyone explain where I am going wrong and how I can reduce it further?