If $H$ and $K$ are solvable subgroups of $G$ with $H\unlhd G $ then $HK$ is a solvable subgroup of $G .$
Here is my proof :
Now if we can show the quotient group $\displaystyle\frac{HK}{H}$ and $H$ are both solvable then so is $HK .$
So now the only task is to prove that this quotient group $~\displaystyle\frac{K}{H\cap K}~$ is actually solvable .
Note that $H\unlhd G$ and $H\subseteq HK $ which implies that $ H\unlhd HK\le G .$ Then by second isomorphism theorem for groups to give
$$\frac{HK}{H}\cong \frac{K}{H\cap K} ~.$$
Then the quotient group $\displaystyle\frac{K}{H\cap K}$ is solvable since it is the homomorphic image of the solvable group $K$ if we consider the natural projection $\varphi:K\longrightarrow\displaystyle\frac{K}{H\cap K}~.$
Is there anyone to check my working for validity ? Any advice will be appreciated . Thanks for patient reading .