1.1. It is known that $f(f(x))=x^{4}+4 x^{3}+8 x^{2}+8 x+4$. Find all such polynomials $f(x)$.
Solution. Let $f(x)=a x^{2}+b x+c$. Then $f(f(x))=a^{3} x^{4}+2 a^{2} b x^{3}+\ldots$ From this, $a=1, b=2$. Finally, let's look at the coefficient of $x^{2}$ and determine that $c=1$. Indeed,

$$
\left(x^{2}+2 x+1\right)^{2}+2\left(x^{2}+2 x+1\right)+1=x^{4}+4 x^{3}+8 x^{2}+8 x+4
$$