Normal form of a vector field in $\mathbb {R}^2$ Edited after considering the comments
Problem: What is the normal form of the vector field:
$$\dot x_1=x_1+x_2^2$$
$$\dot x_2=2x_2+x_1^2$$
Solution: The eugine values of the matrix of the linearised around $(0,0)$ system are $2$ and $1$. We, therefore, have the only resonance $2=2\dot{}1+0\dot{} 2$. The resonant vector-monome is $(0,x_1^2)$. The normal form is then
$$\dot x_1=x_1$$
$$\dot x_2=2x_2+cx_1^2$$
Question: I believe this is correct, is it not?

I would use $y$ instead of $x$ in the normal form, since these are not the same variables. Otherwise, what you did is correct. (I don't know if the problem required the identification of  a transformation between $x$ and $y$.)
