Currently I'm learning about functions in my AoPS book. One of the problems is to find all possible solutions to the equation $$f(x+y)+f(x-y)=2x^2-2y^2$$ The book currently explains two methods of solving functional equations. One method is isolation but I don't think that'll work for this equation and another method is to substitute values in for $y$ which might work.
My approach: I substituted $y=0$ and got $2f(x)=2x^2$ so i get $f(x)=x^2$ but once I plug my results back into the equation I get $2x^2+2y^2$ so this solution is wrong. Now I'm stuck because this is the only approach I have for this problem.
What method should I use to approach this problem and is there even a possible solution?