  Finding an effective formula for describing a discriminant of a quadrinomial
(a formula which can be easily computed for high values of degrees of
quadrinomials) is a difficult problem. In 2018 Otake and Shaska using advanced
matrix operations found an explicit expression of $\Delta(x^n+t(x^2+ax+b))$. In
this paper we focus on deriving similar results, taking advantage of
alternative elementary approach, for quadrinomials of the form $x^n+ax^k+bx+c$,
where $ k \in \{2,3,n-1\}$. Moreover, we make some notes about
$\Delta(x^{2n}+ax^n+bx^l+c)$ such that $n>2l$.
