When does the product of two polynomials = $x^{k}$? Suppose $f$ and $g$ are are two polynomials with complex coefficents (i.e $f,g \in \mathbb{C}[x]$).
Let $m$ be the order of $f$ and let $n$ be the order of $g$.
Are there some general conditions where  
$fg= \alpha x^{n+m}$
for some non-zero $\alpha \in \mathbb{C}$ 

We don't need the strong property of UFD. If $\rm D$ is a domain $\rm D$ then $\rm x$ is prime in $\rm D[x]$ (by $\rm D[x]/x \cong D$ a domain), and products of primes factor uniquely in every domain (same simple proof as in $\Bbb Z$). In particular, the only factorizations of the prime power $\rm x^i$ are $\rm \,x^j x^k,\ i = j+k\ $ (up to associates as usual). This fails  over non-domains, e.g. $\,\rm x = (2x+3)(3x+2) \in \mathbb Z/6[x].$
