If I didn't mess up, this feels easy for it's position. Consider polynomials $\tilde{P}, \tilde{Q}, \tilde{R} \in \mathbb{R}[x, y]$ such that $P(x, y)=\tilde{P}(x-y, x+y)$, and idem for $Q, R$. The expression rewrites conveniently in polynomial form $$\tilde{P}(XU, YV)=\tilde{Q}(X, Y)\tilde{R}(U, V),$$where $$X=x-y, Y=x+y, U=u-v, V=u+v$$are four variables that can freely span $\mathbb{R}^4$. For the product of two monomials $X^{i}Y^j$ and $U^kV^l$ to be a term in $\tilde{P}(XU, YV)$, we require of course that $i=k, v=l$. This means that the only possible choice, which is fine, is $$\tilde{Q}=X^nY^m, \tilde{R}=U^nV^m,$$which finally means that the solutions are of the form $$ P(a, b)=c_1c_2(a-b)^n(a+b)^m ( Q(x, y)=c_1(a-b)^n(a+b)^m, R(a, b)=c_2(a-b)^n(a+b)^m$$where $n, m$ are positive integers, and $c_1, c_2$ real constants. It is clear that these work, as we verified when we extracted the answer.