Suppose that $f'(x)g(x)=f(x)g'(x)$ and $g(x)\ne 0$ on (a,b). How are $f$ and $g$ related?

In your quotient rule, it must be $(-)$. Now, using the assumption, $h^\prime(x)=0$ on $(a,b)$. What can you conclude about $h$ on $(a,b)$?

Remember your definition of h. You have just managed to prove that: $$\frac{d}{dx}\frac{f(x)}{g(x)} = 0$$ This implies: $$f(x) = Cg(x)$$ Where C is any constant.