Existence of primes Of a given form

Question

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Given a prime $p$ and an integer $n \neq 1$, not divisible by $p$, does there always exists a prime, or possibly infinitude of primes $q$ such that $p+nq$ is also prime? Same questions when $P$ is an irreducible polynomial, $N$ is a polynomial which does not divide $p$, and we want a prime number $q$ such that $P(x)+qN(x)$ is irreducible.

Answer 1

For odd $p$ and odd $n$ there need not exist any prime $q$ such that $p+nq$ is prime. Take $p=19$ and $n=3$. Suppose that $p+nq=19+3q$ is prime. This is always even, except for $q=2$, where we have $p+nq=25$.