Polynomials dense with primes

It is known (it follows from Brun's sieve, or more modern sieves) that for any fixed polynomial $p$, there exists a constant $c_p$ such that $$ \# \{ n\le x\colon p(n) \text{ is prime} \} < c_p \frac x{\log x}. $$ In particular, your density $\Delta$ equals $0$ for any polynomial (as Will Sawin commented).

For irreducible polynomials without obvious obstructions (such as all the values being even), it is conjectured that $\# \{ n\le x\colon p(n) \text{ is prime} \} \sim s_p \frac x{\log x}$ for some constant $s_p$ as $x\to\infty$; but this is an open problem for any polynomial $p$ of degree greater than $1$.

This has been asked multiple times before (with three variations by yours truly), so is a mega-duplicate, if you will:

Bateman-Horn, continued even further

Bateman-Horn conjecture, continued

Unexpectedly prime rich cubic polynomial

And even resulted in a preprint:

Some experiments on Bateman-Horn

I Rivin - arXiv preprint arXiv:1508.07821, 2015 - arxiv.org