let $\mathbb{P}_{n,m}$ be a set of polynomials $P(x,s)$ with complex coefficients such that $P(x,s) = 0$ or $deg(P(x,1)) \leq n-1 $ and $deg(P(1,s)) \leq m-1$ show that $\phi: \mathbb{P}_n \otimes \mathbb{P}_m \to \mathbb{P}_{n,m}$ defined by $\phi(P(x) \otimes Q(s)) = P(x)Q(s) $ is a isomorphism.
I ready prove that $\phi$ is morphism, but 1-1 and surjective is not simple for me!!
$\mathbb{P}_m$ is spanned as a vector space by $1,s,\dots,s^{m-1}$, thus you may write any (nonzero) element of $\mathbb{P}_n\otimes\mathbb{P}_m$ as $$ \sum_{i=0}^{m-1}f_i(x)\otimes s^i. $$ You should be able to derive injectivity from this.
For surjectivity, play the same game. Take any polynomial $P(x,s)$ and write it $$ P(x,s)=f_0(x)+f_1(x)s+\dots+f_{m-1}(x)s^{m-1}. $$ This is obviously in the image of your map based on the discussion above.