302. Let the polynomials $f(x)$ and $g(y)$ have free terms $a_{0}$ (i.e., $f(x) \equiv a_{0} + a_{1} x + \ldots + a_{n} x^{n}$) and $b_{0}$. Set the variable $x$ to 0 in the equation $x^{200} y^{200} + 1 \equiv f(x) g(y)$; then we get $a_{0} g(y) \equiv 1$, i.e., $g(y) = \frac{1}{a_{0}}$; thus, $g(y)$ equals $\frac{1}{a_{0}}$ for all $y$, i.e., it is a constant $\frac{1}{a_{0}}$ (a polynomial of degree zero). Similarly, it can be shown that $f(x) = \frac{1}{b_{0}}$, i.e., $f(x) g(y) = \frac{1}{a_{0} b_{0}} \neq x^{200} y^{200} + 1$. The obtained contradiction proves the statement of the problem.