The following is a homework question, I'm just looking for guidance as to what I can do to approach the question (it isn't for any credit, but homework nonetheless). I am stumped on how to approach it. Any tips or hints would be greatly appreciated, thank you!
Do polynomials $a(x), b(x) \in R[x]$ and $c(y), d(y) \in R[y]$ exist such that $$1+xy+x^2 y^2 = a(x)c(y) + b(x)d(y)$$
No. There are no such polynomials.
Note that $ 0 \neq \frac{3}{4} + \left( xy + \frac{1}{2} \right)^{2} = 1+ xy + x^{2}y^{2} = \det A (x, y)$, where $A(x, y) = \begin{pmatrix}a(x) & -d(y) \\ b(x) & c(y) \\ \end{pmatrix} $
This means that the columns of $A(x, y)$ are linearly independent for all $x$ and for all $y$.
If you now suppose such polynomials exist, then plugging in $y = 0$ and then again by plugging in $x = 0$ into the original equation, we see that $a(x)$ and $b(x)$ cannot have a common root as that leads to 1 =0. Similarly, for $c(y)$ and $d(y)$. Suppose $\alpha$ and $\beta$ are roots of $b(x)$ and $c(y)$ respectively, then $\det A (\alpha, \beta) =0$, which is a contradiction.