Can we find linear factors for the following polynomial with two variables? $$1+x+y-(x+y)^2$$
My Try
First I tried writing two linear factors in general, $$1+x+y-(x+y)^2=(Ax+By+C)(-\frac{1}{A}x-\frac{1}{B}y+\frac{1}{C})$$ During comparison of coefficients I ran into some trouble.
comparing xy terms:$ A=B$--(1)
comparing x terms: $AC=A^2-C^2$ --(2)
comparing y terms: $CB=B^2-C^2$--(3)
From (3)-(2) I get, $(B-A)(B+A)=(B-A)C$
from this point can I cancel (B-A) term from both sides? ( My guess is we can't because A=B) so how should I proceed? Thanks in advance!
Your idea is not bad. We can restate the factorization with $a=b$ and we can also set WLOG $c=1$ (because $a,b,c$ are proportional). This gives
$$(ax+ay+1)\left(-\frac xa-\frac ya+1\right)=1+\left(a-\dfrac1a\right)x+\left(a-\dfrac1a\right)y-x^2-2xy-y^2=0.$$
Now it suffices to solve
$$a-\dfrac1a=1.$$
Anyway, it is much easier to factor $1+t-t^2$, as suggested by @Ivan.