let $f(x)$ is polynomial function , $f(x+2)-f(x) = 6x^2+12x$ , $f(1)-f(0)=?$

Question

Let $f$ is polynomial function.

$$f(x+2)-f(x) = 6x^2 +12x$$

$$f(1)-f(0)=?$$

I found that $f(x)$ is 3rd order ,

and use some coefficient $a, b, c, d$, but that's too hard to solve.

I want to use difference equation /recurrence or any smart way.

Answer 1

Evaluating the functional equation at $x=0$ and $x=-2$ gives that $$f(2)-f(0)=0\\f(0)-f(-2)=0$$ Thus: $$f(-2)=f(0)=f(2)=a$$ This gives us that $f(x)=g(x)*x(x-2)(x+2)+a$.

Substituting $f(x)$ back into the initial equation gives $g(x)=1$ and hence $f(1)-f(0)=-3$