Implicit polynomial

There is a function $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(x)+x^2 (f(x))^3+x+(f(x))^5x^4=1$ for all $x\in\mathbb{R}$ . What can we say about $f(x)$ ?

Bonus : If possible, find the third order Taylor series of $f(x)$ around $x=0$ .