Given the ideal $I= \langle 3x^2 + y^2, xy \rangle \subset \mathbb{C}[x,y]$ show that $\mathbb{C}[x,y]/ \langle 3x^2 + y^2, xy \rangle= \mathrm{span}_\mathbb{C}\{1,x,y,x^2 \}$.
I understand how to do a simpler problem, say if the ideal was $I= \langle x,y \rangle$ then it makes sense to me why $ \mathbb{C}[x,y]/ \langle x,y \rangle =\mathrm{span}_\mathbb{C} \{1 \}$ but when I try work through the problem with a more complicated ideal I cannot seem to make sense of it, particularly the $x^2$ confuses me. I’ve seen questions with similar examples on here but none explain the answer explicitly.
$$\Bbb{C}[x,y]/(3x^2+y^2) = \Bbb{C}[x]+y\Bbb{C}[x]$$
$$\Bbb{C}[x,y]/(3x^2+y^2,xy^2)=\Bbb{C}[x,y]/(3x^2+y^2,x^3)=\Bbb{C}[x]/(x^3)+y\Bbb{C}[x]/(x^3)$$
$$\Bbb{C}[x,y]/(3x^2+y^2,xy^2)/(xy)=(\Bbb{C}[x]/(x^3)+y\Bbb{C}[x]/(x^3)) / (xy)= \Bbb{C}+x\Bbb{C}+x^2\Bbb{C}+y\Bbb{C}$$