It seems to me that this is conjectured by Schinzel. His conjecture for this special case is as follows:

**Conjecture(Schinzel 1995):** For every algebraic number field $K$ there exists a constant $C_1(K)$ such that, if $n > 2$, $A,B ∈ K^{*}$ and the trinomial $x^n +Ax +B$ is reducible over $K$ then either it has a proper factor of degree $≤ 2$ or $n ≤ C_1(K)$. (For $K = \mathbb{Q}$ we have $C_1(\mathbb{Q}) ≥ 52$.)

In other words, we should find an integer $n\ge C_1(K)$, such that $x^n+x+B$ is irreducible. The reference is: A. Schinzel: "Solved and unsolved problems on polynomials", $1995$.