If $f(x+1)+f(x-1)=4x^2-2x+10$ then what is $f(x)$ 
If $$f(x+1) +f(x-1)= 4x^2 -2x +10$$  then what is $f(x)$ 

What is strategy of solving this kind of  problems ?
Thank you for help

Using Taylor's formula for polynomials:


*

*$f(x+1)=f(x)+f'(x)\cdot 1+f''(x)\cdot\dfrac 12$,

*$f(x-1)=f(x)-f'(x)\cdot 1+f''(x)\cdot\dfrac 12$.


Therefore $\;f(x+1)+f(x-1)=2f(x)+f''(x)=4x^2-2x+10$. If we set $f(x)=ax^2+bx+c$, this equation simplifies to
$$ax^2+bx+c+a=2x^2-x+5,\enspace\text{whence}\quad a=2, b=-1, c=3.$$
