prove big-O relationship

To prove $f(n)=O(g(n))$, find some $M$ and $c$ and prove that for all $n\ge M$, we have $c\cdot g(n)\ge f(n)$.

A simple way to prove this is to add the coefficients of all terms of $f(n)$ together, and multiply $g(n)$ by that. For example $f(n)=2n^2+2n+2=O(n^2)$ because we can see that $2n^2+2n+2\le 2n^2+2n^2+2n^2=6n^2=c\cdot n^2$. Now we just need to find the $M$ for which this inequality holds for all $n>M$.