Problem: Use polynomial long division to find the extreme point to

$\displaystyle f\left( x,y \right)=\left( x^{2}+y^{2} \right)\left( xy+1 \right)=x^{3}y+x^{2}+xy^{3}+y^{2}$

Attempt:

$\displaystyle f\left( x,y \right)=x^{3}y+x^{2}+xy^{3}+y^{2}$

$\displaystyle \frac{\partial f}{\partial x}=3x^{2}y+2x+y^{3}$

$\displaystyle \frac{\partial f}{\partial y}=x^{3}+3xy^{2}+2y$

I see that (0,0) is an extreme point, but I havn't learned how to use polynomial long division with more than one variable. How do I do it when I have both x and y?