How to determine if $2+x+y$ is a factor of $4-(x+y)^2$? I know it is a factor but how could have I determined that it was? Feel free to link whatever concept is needed than solve it. Studying for clep and it's one of the practice problems. When I expand it I get nonsense.

The solution's already in the other answers, but in many cases, as in this one, you can try some substitution:
$$t:=x+y\implies\;\text{is}\;\;2+t\;\;\text{a factor of}\;\;4-t^2\;?$$
and now all depends on you remembering the high school algebra's slick formula, namely difference of squares:
$$4-t^2=(2^2)-t^2=(2-t)(2+t)\;\;\text{and etc.}$$
