## Asymptotic analysis

Which of the following conjecture is true? Justify

$10n = O(n)$ $10n^2 = O(n)$ $10n^{55} = O(n^2 )$

1) $10n \leq cn$ => $n=1$ $c=10$

$f(n) \leq 10g(n)$ for all $n \geq 1$. $f(n) \in O(g(n))$

How can I solve the other two?

I found this problem on the Internet:
It is true that $n^2 + 200n + 300 = O(n^2 )$ ? And $n^2 -200n -300 = O(n)$ ?