Let $P(x) $ be a polynomial of degree $n\ge 4$ with integer coefficients and constant term equal to $1$. I am interested in Polynomials $P(x) $ such that for a fixed positive integer $b$, there are finitely many positive integers $x$ such that $P(x) $ contains a proper divisor $d \equiv1\pmod x$ and $P(x+b) $ contains a proper divisor $d' \equiv1\pmod {x+b} $.
I know addition and multiplication generally do not see each other but out of curiosity, are there any such polynomials $P(x) $ with this property? Take $P(x) = x^4+x^3 +x^2 +x+1$ and $b=2$. After a long search through all $x < 10 ^5$, I find that only $x = 102$ satisfies this condition ($ P(102) =(15\cdot 102 +1)(700 \cdot 102+1)$ and $ P(104) =(5\cdot 104+1)(2180 \cdot 104+1)$).