Since, $N \unlhd G$ and $H \le G$, from the second isomorphism theorem we know that $N \unlhd NH$, therefore $NH$ is solvable if $N$ and $NH/N$ are solvable.
From our assumptions, $N$ is solvable, therefore the question is whether $NH/N$ is solvable or not.
Again, from the second isomorphism theorem, we know that $ NH/N \simeq H/ H \cap N$.
Since, $H$ is solvable and $H\cap N \unlhd H$, $H/ H \cap N$ must be solvable, therefore $NH/N$ is solvable and so is $NH$.
Is this correct?