Affine Non-Reductive GIT and moduli of representations of quivers with multiplicities

We give an explicit approach to quotienting affine varieties by linear actions of linear algebraic groups with graded unipotent radical, using results from projective Non-Reductive GIT. Our quotients come with explicit projective completions, whose boundaries we interpret in terms of the original action. As an application, we construct moduli spaces of semistable representations of quivers with multiplicities subject to certain conditions, which always hold in the toric case for a generic stability condition.