I'm trying to prove this ideal
$$I=(x^2+y^2+x,x+y+xy)\subset \mathbb C[x,y]$$
I supposed that $I$ is prime and I'm using the classical method to prove $I$ is prime: If $ab\in I$, then $a\in I$ or $b\in I$. I didn't have any success with this method and I would like to know which are the tools and strategies I could use to solve this question.
I really need help.
Thanks in advance.