Given a poly(log(G), log(x)) algorithm and N>G, is it also poly(log(N), log(x))?

Question

I have constructed an algorithm that runs in poly(log(G), log(x)). I am now asked to construct one that runs in poly(log(N), log(x)), where N>G. What would make the first algorithm not also run in poly(log(N), log(x))? (is the fact that N>G not enough?).

Thanks.