Functional equation with two variables: $ f(x) - f(y) = (x+y)(x-y) $

Question

Find all functions that, for all real numbers $ x $ and $ y $, satisfy the following functional equation:

$$ f(x) - f(y) = (x+y)(x-y) $$

Answer 1

Hint: $f(x) - x^2 = f(y) - y^2$, so this must be ...

Answer 2

Putting $y=0$, we get $f(x)−f(0)= (x+0)(x-0) = x^2$ and hence $f(x)=f(0)+x^2$. The function is of the form $f(x)=k+x^2$ for any constant $k$. There are infinitely many functions satisfying the given condition.

Answer 3

$$f(x)-f(y)=(x+y)(x-y)$$

so: $$f(x)-f(y)=x^2-y^2$$ and: $$f(x)-x^2=f(y)-y^2$$ Left Side has only $x$ and Right Side has $y$ argument and each side is equal to another, so both sides must be equal to a Constant, like $K^2$.

so: $$f(x)-x^2=f(y)-y^2 = K^2$$ and: $$ f(x)=x^2 + K^2\\ f(y)=y^2 + K^2 $$