  In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities
on general stratified groups and weighted Hardy inequalities on general
homogeneous groups using the factorization method of differential operators,
inspired by the recent work of Gesztesy and Littlejohn \cite{GL17}. We note
that some of the obtained inequalities are new also in the usual Euclidean
setting. We also obtain analogues of Gesztesy and Littlejohn's 2-parameter
version of the Rellich inequality on stratified groups and on the Heisenberg
group, and a new two-parameter estimate on $\mathbb{R}^{n}$ which can be
regarded as a counterpart to the Gesztesy and Littlejohn's estimate.
