Run time of product of polynomially bounded numbers
Computing the product sequentially, we need to perform $n-1$ multiplications of numbers of size $O(C^n n^{cn})$, for some constant $C$ corresponding to the constant in $O(n^c)$. Using algorithms for fast integer multiplication, each of these requires $\tilde{O}(C\log n + cn \log n) = \tilde{O}_c(n)$, so in total the time is $\tilde{O}_c(n^2)$ (the $\tilde{O}$ hides logarithmic factors).