Given $F(N) = 55N(600 + 50N \log N + 20N) + 20N(30N + 20\sqrt N)(50 + \log N)$.
How can one combine multiplication with addition for a Big-O estimate by algebraic means? I'm used to simply taking the bigger terms of simpler expressions as the Big-O (i.e. $10 + 3N^2 + 20 = O(N^2)$).
Just expand it. $$F(N) = a_1N + a_2N^2\log N + a_3N^2 + (a_4N^2 + a_5N\sqrt N)(a_6+\log N)$$ $$F(N) = a_1N + a_9N^2\log N + a_7N^2 + a_8N\sqrt N+a_5N\sqrt N\log N$$ $$F\in O(N^2\log N)$$
Note that I've replaced all the constants with arbitrary variables, since they don't matter anyways.