Generalizing two families of scattered quadrinomials in $\mathbb{F}_{q^{2t}}[X]$

In recent years, several efforts have focused on identifying new families of scattered polynomials. Currently, only three families in $\mathbb{F}_{q^n}[X]$ are known to exist for infinitely many values of $n$ and $q$: (i) pseudoregulus-type monomials, (ii) Lunardon-Polverino-type binomials, and (iii) a family of quadrinomials studied in a series of papers. In this work, we provide sufficient conditions under which these quadrinomials, denoted by $ψ_{m,h,s}$, are scattered. Our results both include and generalize those obtained in previous studies. We also investigate the equivalences between the previously known families of scattered polynomials and those in this new class.