*After several useful comments, I understood that I meant two questions rather than one. Sorry for messing it up, and feel free to answer any of them or vote for closure!*

If $G$ is a connected, solvable affine algebraic group (over $\mathbb C$) and $H$ is a closed subgroup, is $G/H$ an affine variety? It is th. 4.3, p. 184 of Hochschild's

*Basic theory of algebraic groups and Lie algebras*, thanks to Francois Ziegler's comment.If $G$ is a connected, unipotent affine algebraic group and $H$ is a connected closed subgroup, is $G/H$ an affine space? As $G=\mathbb A^n$ and $H=\mathbb A^k$ as varieties, intuitively, it is a question whether $\mathbb A^n / \mathbb A^k = \mathbb A^{n-k}$. It is true if $H$ is normal, because $G/H$ is connected, unipotent (consists of unipotent elements), and also affine by Ziegler's comment.

For a normal $H$, the second question can also be proven directly: the orbits of a unipotent group $H$ on an affine variety $G$ are closed, therefore the affine quotient $\mathrm{Spec} \; \mathbb C[G]^H$ (where, a priori, the ring is not finitely generated) is geometrical and coincides with $G/H$.

*Old version of the question:*

If I am not wrong, any flag variety $G/P$ (where $G$ is semi-simple, $P$ is parabolic) is covered by the orbits of a Borel subgroup $B$, each of which is isomorphic to $\mathbb A^k$ for some $k$.

So I wonder whether every quotient of a connected solvable affine algebraic group by a connected but not necessarily normal subgroup is an affine space?

If not, maybe it is an affine or quasi-affine, variety, or what are the conditions that it is? My motivation is why any spherical variety (that is, a $G$-variety with an open $B$-orbit) is covered by open $B$-stable affine subsets.