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?