<s>[INST] Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$. 
Prove that $m=(x+y^2, y+x^2+2xy^2+y^4)$ is a maximal ideal of $\mathbb{C}[x,y]$.

I can show that the ideal $(x,y)$ of $\mathbb{C}[x,y]$ contains $m$ and $(x,y)$ is a maximal ideal. Therefore to show that $m$ is itself a maximal ideal I only need to show that $m=(x,y)$. I tried to prove it by looking for two polynomials $f,g$ such that $x=f(x,y)(x+y^2)+g(x,y)(y+x^2+2xy^2+y^4)$ but wasn't able to find them. And I'm not sure whether this is the best way of proving this. I would appreciate any help. Thank you.
 [/INST] Indeed, $x=\color{red}{f(x,y)}(x+y^2)+\color{green}{g(x,y)}(y+x^2+2xy^2+y^4)$ with, for example, $\color{red}{f(x,y)}=1-(x+y^2)^3$ and $\color{green}{g(x,y)}=(x+y^2)^2-y$.
This can be found by noting that $a=x+y^2$ and $b=y+x^2+2xy^2+y^4$ are such that $b=y+a^2$ hence $x=a-y^2=a-(b-a^2)^2$. Thus, $x=\color{red}{(1-a^3)}a+\color{green}{(2a^2-b)}b$.
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