Okay, here is the idea. Given a positive integer N, we want to find polynomials P(x), Q(x), with integer coefficients, each having height at least N, such that P(x)Q(x) has height 1.

Even for N = 2 this takes a bit of thought. Start with the polynomial

which clearly has height 1. Now by repeatedly factorising the difference of two squares, write that polynomial as

Take

and

Then each of P(x), Q(x) has height at least 2, and their product has height 1.

Now do the same sort of thing for N = 3, looking at the polynomial

Once you have sorted that one out, you will see how to do it for any N. (It's really a very simple-minded construction, nothing as sophisticated as cyclotomic polynomials!)