How to prove polynomials with degree $n$ does not form a vector space?

Polynomials of degree $n$ does not form a vector space because they don't form a set closed under addition.

For instance:

$$X^n-X^n=0$$

which is not of degree $n$.

So, don't get confused with the set of polynomials of degree less or equal then $n$, which form a vector space of dimension $n+1$. We often work with this space.

Polynomials of degree $n$ is a set which is not closed under addition. For example, if $n=3$, then $x^3+x^2$ and $-x^3$ are both $3$rd degree polynomials but their sum is not: $$ x^3+x^2-x^3=x^2 $$ (which is not a $3$rd degree polynomial).