Math Is Fun Forum

Hello, I would like to ask for a help. I am suppose to calculate the Grobner basis of this:

My solution: I think

and I have:

Now I have in the paper the next step and there is writted that -x is f_3 because

I calculated:

but in this moment, I really really do not know, why

    - I think it is not. It is definetelly not, because we can not divide a polynomial with a lower degree by a polynomial with a higher degree. I'm stuck at this point and can't continue, please help. Why

and why 

    ? Help please, thank you.