Does $f(n) + g(n) = O(g(n) \cdot f(n))$ hold?
I don't get what you do with your choice of $f$ and $g$ but they don't work as a counter-example.

Here is how you see that:

$\qquad f(n) + g(n) = 3n^2 + n \in \Theta(n^2) \subseteq O(n^3) = O(f \cdot g)$.

See our reference question on how to show this.

It is easy to see that the same happens for all pairs of increasing functions: 

$\qquad f+g \in \Theta(\max(f,g)) \subseteq O(f \cdot g)$.

So if you want to have a chance at disproving the claim, you'll have to look for decreasing functions -- if your definition of Landau symbols admits such.