If $p:\mathbb{R}^n\to \mathbb{C}$ is a polynomial, show that there exist constants $C>0$, $R>0$, and $N\in\mathbb{N}$ such that $$ |p(x)|\leq C(1+||x||^2)^N, \;\; \text{for} \;||x|| \geq R $$ is valid for all $x\in\mathbb{R}^n$.
My attempt so far:
Let $N\in\mathbb{N}$ sufficiently larger with respect to $\partial(p)$.
Then, there exists a $R>0$ such that for $||x||\geq R$ we have $$\frac{|p(x)|}{(1+||x||^2)^N} < 1$$.
Let $M = \max\left\{\frac{|p(x)|}{(1+||x||^2)^N}:x\in B(0,R)|\right\}$.
It is enough to define $C>0$ such that $C= \max\{M,1\}$.
Is this approach correct?