Find polynomial in equation Find polynomial $f(x)$ with real coefficients that satisfied:
$$x^2f(x) + 2 = f(x^2) + 2x^3$$
I find that $\deg f$ can be $1$ and $2$.
$$\deg f = n $$
$$2+n=\max(2n,3)$$


*

*First case $2=n$

*Second case $n=1$
But what next? 

Your polynomial has the form $f(x)=ax^2+bx+c$ with $a,b,c\in \mathbb{R}$.
Thus your equation gives
$$ax^4+bx^3+cx^2+2=ax^4+bx^2+c+2x^3$$and comparing the terms of each degree on both sides yields $b=c=2$, and $a$ can be any real number.
