Summary: Let \(f(x)\in\mathbb{Z}[x]\) be monic and irreducible over \(\mathbb{Q}\), with \(\deg (f)=n\). Let \(K = \mathbb{Q}(\theta)\), where \(f(\theta)=0\), and let \(\mathbb{Z}_K\) denote the ring of integers of \(K\). We say \(f(x)\) is monogenic if \(\{1,\theta,\theta^2,\dots,\theta^{n-1}\}\) is a basis for \(\mathbb{Z}_K \). Otherwise, \(f(x)\) is called non-monogenic. In this article, we give necessary and sufficient conditions for a certain class of polynomials to be monogenic. Using these conditions allows us to generate infinite families of non-monogenic polynomials. In particular, for quadrinomials our results show that there exist infinitely many primes \(p\geq 3\), and integers \(t\geq 1\) coprime to \(p\), such that \(f(x)=x^p-2ptx^{p -1}+p^2t^2x^{p-2}+1\) is non-monogenic. Finally, we illustrate this situation with an explicit example.