Find the product of $f(x)$ and $g(x)$ given one of them I'm given the final answer which is $$(g \cdot f)(x) = \frac{1}{x^2+4}\;.$$
Also, i'm given $f(x) = x^2+1$.
I've solved this using the composition, however the second part of the question asks me to find the $g(x)$ which would make this multiplication true. How would I do this? Do I divide the final answer by $f(x)$? 

Yes, becasue by definition $(g\cdot f)(x)=g(x)\cdot f(x)$.
Hence $g(x)=\frac1{(x^4+1)(x^2+1)}$. 
If we had $(g\circ f)(x)=\frac1{x^4+1}$ instead, one possible $g$ would be $g(x)=\frac1{(x-1)^2+4}$.
