  We study quotients of quasi-affine schemes by unipotent groups over fields of
characteristic 0. To do this, we introduce a notion of stability which allows
us to characterize exactly when a principal bundle quotient exists and,
together with a cohomological vanishing criterion, to characterize whether or
not the resulting quasi-affine quotient scheme is affine. We completely analyze
the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here
the above characterizations admit simple geometric and algebraic
interpretations. As an application, we produce arbitrary dimensional families
of non-isomorphic smooth quasi-affine but not affine n-dimensional varieties (n
\geq 6) that are contractible in the sense of A^1-homotopy theory. Indeed,
existence follows without any computation; yet explicit defining equations for
the varieties depend only on knowing some linear G_a- and SL_2- invariants,
which, for a sufficiently large class, we provide. Similarly, we produce
infinitely many non-isomorphic examples in dimensions 4 and 5. Over C, the
analytic spaces underlying these varieties are non-isomorphic, non-Stein,
topologically contractible and often diffeomorphic to C^n.
