Answer is no. Let $P_1,P_2$ be trinomials. Suppose the contrary that for some positive integer $a$, $x+a$ is a factor of $P_1,P_2$ and $k\in\mathbb{N}$ is such that $P_1(k)=19$ and $P_2(k)=98$. So, $k+a\mid {\rm gcd}(19,98)=1$. As $k+a\ge 2$, this is a contradiction.