I've been wondering recently about results for irreducibility that use the "additive structure" of the polynomial ring at hand. For instance, can we say anything about the irreducibility of a sum of two irreducible polynomials that satisfy some conditions? This seems like an additive number theory problem but in a different context. I searched the literature for stuff along these lines but did not find much. Am I missing something obvious (like maybe I can be asking the same questions in the underlying ring) or is this just a question with sparse literature around it? Any papers involving similar questions would be extremely appreciated!

Here's an example: $(x^{2}+2x+1)p(x)+x$ seems to be irreducible for "almost all" p(x) with nonzero constant term. Unsure how I'd show this. Here's a paper that kind of has the flavor of problems that I'm interested in.