  We improve Irving's method of the double-sieve by using the DHR sieve. By
extending the upper and lower sieve functions into their respective
non-elementary ranges, we are able to make improvements on the previous records
on the number of prime factors of irreducible polynomials at prime arguments.
In particular, we prove that irreducible quadratics over $\mathbb{Z}$
satisfying necessary local conditions are $P_4$ infinitely often.
