Some observations concerning reducibility of quadrinomials

In a recent paper [4], Jankauskas proved some interesting results concerning the reducibility of quadrinomials of the form f (4, x), where f(a,x)=xⁿ+x^m+x^k+a. He also obtained some examples of reducible quadrinomials f (a, x) with a∈Z, such that all the irreducible factors of f (a, x) are of degree ≧3. In this paper we perform a more systematic approach to the problem and ask about reducibility of f (a, x) with a∈Q. In particular by computing the set of rational points on some genus two curves we characterize in several cases all quadrinomials f (a, x) with degree ≦6 and divisible by a quadratic polynomial. We also give further examples of reducible f(a, x),a∈Q, such that all irreducible factors are of degree ≧3.