I want to show that the polynomial
$$f(x)= x^4 -2x^2 +8 x+1$$
is irreducible over $\Bbb Q$.

I've proved it by a long method, but I need an easy and short method.
I've try to put $x=t+1$, but this fails.

**My proof** I reduce $f$ $\bmod 3$. I proved that there is no linear root by showing that $f(0), f(1), f(2)$ nonzero.

I proved that there is no quadratic root by checking all possible products of quadratics $x^ 2 + ax +b $.