Solving recurrence relations with a squared term?

Question

$F_{n+1} = F_n^2+2F_n$. Is there a way to solve this equation with the standard technique of solving the associated quadratic? In general, when can I and when can I not use the "polynomial solution" to a recurrence equation.

For anyone wondering where this recurrence comes from see 1985 Putnam A3.

Answer 1

Hint:

$$F_{n+1}+1=(F_n+1)^2\\\implies F_n+1=(F_0+1)^{2^n}$$

Answer 2

$$F_{n+1}+1=(F_n+1)^2$$ and from here we can get, which you wish.