Question

is x^2-4x-8 prime?

Answer 1

It is prime over the integers, but not over the rationals

Answer 2

You can factor it over the rationals as
\[
x^2-4 x-8=\left(x-2
\left(1-\sqrt{3}\right)\right) \left(x-2
\left(1+\sqrt{3}\right)\right)
\]