Calculate the multiplicity at the origin of the intersection of the curves $X=V(xy^2+x^4)$ and $Y=V(x^2y-x^4)$.
I have tried to do the following:
$\mu_0(xy^2+x^4, x^2y-x^4)\\ =\mu_0(x(y^2+x^3), x^2y-x^4)\\ =\mu_0(x, x^2y-x^4)+\mu_0(y^2+x^3, x^2y-x^4)\\ =\mu_0(x, x^2y)+\mu_0(y^2+x^3, x^2(y-x^2))\\ =\mu_0(x, x^2)+\mu_0(x, y)+\mu_0(y^2+x^3, x^2)+\mu_0(y^2+x^3, y-x^2)\\ =\mu_0(x, x^2)+1+\mu_0(x^2, y^2)+\mu_0(y^2+x^3, y-x^2)\\ =5+\mu_0(x, x^2)+\mu_0(y^2+x^3, y-x^2)$
I don't know how to calculate $\mu_0(x, x^2)$ and $\mu_0(y^2+x^3, y-x^2)$, could someone please help me? Thank you.