solve the functional equation $f(x+t)-f(x-t)=4xt$ I think this question might be related with arbitrary functions, but I’m not sure. I also tried to set $t$ to different values but couldn’t get it to work.
I tried to set $t=x$ and end up with $f(2x)=f(0)+4x^2$, $f(x)=f(0)+x^2$.

Set $x=t$ and we get $$f(2x)-f(0)= 4x^2$$ so $f(x) =x^2+a$ where $a= f(0)$.
Check: If we now put this in to starting equation we get: $$ (x+t)^2+a-(x-t)^2-a = 4xt$$ which is always true. So $\boxed{f(x) =x^2+a}$ for all real $a$.
