Let $(X_t)$ be a process with independent increments such that $X_0=0$ and $E(X_t)=0$
Let $F_t$ be a natural filtration of $X_t$
Let $a$ and $b$ be arbitrary real numbers and let $(Y_t)$ be a random process: $$Y_t:=aX_t^2+bX_t+f(t)$$
Find function $f$ : $[0, \infty)\rightarrow\mathbb{R}$ such that $(Y_t)$ is a martingale with respect to filtration $F_t$
My attempt
Let $t\ge s>0 $ $$E((aX_t^2+bX_t+f(t))|F_s)=aE(X_t^2|F_s)+bE(X_t|F_s)+f(t)=aE((X_t-X_s+X_s)^2|F_s)+bE((X_t-X_s+X_s)|F_s)+f(t)=a(E((X_t-X_s)^2|F_s)+2E((X_t-X_s)X_s|F_s)+E(X_s^2|F_s))+b(E(X_t-X_s|F_s)+E(X_s|F_s))+f(t)=aE(X_t-X_s)^2+aX_s^2+bX_s+f(t)$$
So we have that the following equality must hold: $$aE(X_t-X_s)^2+f(t)=f(s)$$ so $$f(t)=f(s)-aE(X_t-X_s)^2=f(s)-aVar(X_t)+aVar(X_s)$$ for $t\ge s>0 $
I can't see how I can go further :/ Any help?
$$f(t)=-aE(X_t^2){}{}{}{}{}{}$$