Find all real coefficient polynomial such $f(x)|f(x^2-2)$ 
Find all real coefficient polynomial with the degree is $3$,and such
  $$f(x)|f(x^2-2),\forall x\in Z $$

I try: let $f(x)=ax^3+bx^2+cx+d$,then $$f(x^2-2)=ax^6-6ax^4+12ax^2-8a+bx^4-4bx^2+4b+cx^2-2c+d$$,then such 
$$(ax^3+bx^2+cx+d)|ax^6-6ax^4+12ax^2-8a+bx^4-4bx^2+4b+cx^2-2c+d,\forall x\in Z $$following I can't it 

Hint: Write $$f(x^2-2)=g(x)f(x)$$ If $x$ is a root of $f$ then $x^2-2$ is a root too. Then $(x^2-2)^2-2$ is a root of $f$ and etc. But $f$ cannot have infinitely many roots.
