Is $x^2+25x+4 \in \mathcal{O}(x^2)$? If yes how? If no why not?

Question

Is $x^2+25x+4 \in \mathcal{O}(x^2)$ ? if yes how ?, if no why?

I know $x^2+25x+4\leq 25x^2+25x+25\leq 25x^2+25x^2+25x^2=75x^2$ for some $x$.

What confuses me is $x^2+25x+4\leq 25x^3+25x+25\leq 25x^3+25x^3+25x^3=75x^3$ for some $x$.

Why is the expression $O(x^2)$? Why is it not $O(x^3)$?

Answer 1

The easiest way (denoting your expression $f(x)$): for some $x$ (which?) $$ f(x) \leq 75 x^2 = O(x^2) $$

EDIT: clearly $x^2 +25 x + 4 \leq 25 x^2 + 25 x + 25 \leq 25 x^2 + 25 x^2 + 25 x^2 = 75 x^2 \ \text{for} \textit{ some } \ x$

Answer 2

I guess that you mean that $x^2+2x+4$ is $\mathcal O(x^2)$ at infinity and for this end it is sufficient to prove that $$\frac{x^2+2x+4}{x^2}$$ has a finite limit at infinity which's easy to show.