Polynomials that form $1+xy+x^2 y^2$ Show that there is no polynomials $a(x), b(x) \in R[x]$ and $c(y), d(y) \in R[y]$ such that $1+xy +x^2 y^2 = a(x) c(y) + b(x) d(y) $

Expanding on @MikeDaas's comment we have$$\begin{align}1&=a(x)c(0)+b(x)d(0),\\1+x+x^2&=a(x)c(1)+b(x)d(1),\\1-x+x^2&=a(x)c(-1)+b(x)d(-1)\\\implies x&=\frac{c(1)-c(-1)}{2}a(x)+\frac{d(1)-d(-1)}{2}b(x),\\x^2&=\frac{c(1)-2c(0)+c(-1)}{2}a(x)+\frac{d(1)-2d(0)+d(-1)}{2}b(x).\end{align}$$Since $1,\,x,\,x^2$ are all linear combinations of $a,\,b$, these two functions span a $3$-dimensional space, a contradiction.
