Can somebody please check if the follwoing argument is correct:
Show that is irrdeucible over rationals.
By the rational root theorem it has no rational roots.
Assume that
then
implies that
c=-4 or b-d=-4.
Because bd=-1 it must hold that b=1,d=-1 or b=-1, d=1.
c=-4 shows that d+b=16 which is a contradiction.
if b-d=-4 we also get a contradiction. So the polynomial is irreducible.
Is it correct?