Existence of an irreducible trinomial over finite fields?

Question

Let $F_q$ be the finite field of $q$ elements. I am looking for the existence of an irreducible trinomial of the following form: $$x^n-x^m-1$$ over $F_q$ for some $n,m.$ I think it should be true because we may choose $n$ big enough. In the case, $n$ is fixed, then the conjecture may be false. For example, at $n=2,q=5$, there is no such irreducible polynomial. What do you think?