Let $G$ be the group of all real matrices of the form $ \begin{bmatrix}a&b\\0&c\end{bmatrix} $ with $ac \ne 0$, under matrix multiplication. Let $H$ be the subgroup consisting of all the elements in which $a=c=1$. Use the Fundamental Theorem to show that $ G/H \cong (\Bbb R \setminus \{0\},*) \times (\Bbb R \setminus \{0\},*)$.
It isn't that I want the answer. I'm looking for a certain strategy to develop when facing questions of the form "$G/H \cong something$".
The typical strategy is to leverage the first isomorphism theorem by finding a homomorphism of $G$ onto the second group. I guess this is the 'fundamental theorem' you're referring to.
After picking such a map, you can compute the kernel and see what $H$ must be. Here, in this case, you're given what the kernel is, and that suggests what the homomorphism ought to be. If you map $\begin{bmatrix}a&b\\0&c\end{bmatrix}\mapsto (a,c)$, then $H$ would indeed be the kernel of that map. The only thing to ensure the candidate map you've chosen is actually a group homomorphism.