Do infinitely many prime numbers occur in $P(x) \in \mathbb{N}[x]$ when the coefficients are relative prime?

Question

I have read about Dirichlet's theorem recently, that is, for relative prime positive integers $a,b$, there exists infinitely primes with the form $ax+b$.

What I want to ask is the situation when the $ax+b $ is changed as any irreducible polynomial with relative prime positive integer coefficients. Is there still infinitely many primes?

Answer 1

No, $x^2+2x+1=(x+1)^2$ obviously is not prime for natural $x>0$.

Edit: for irreducible case you may want to read about the (unsolved) Bunyakovsky conjecture.