Let $D(X,Y), E(X,Y)\in\mathbb{Z}[X,Y]$ forms of the same degree $n$ and suppose that the resultant $R=Res(D,E)$ of $D$ and $E$ is not $0$.
Show that there are homogenous forms $L_0(X,Y),M_0(X,Y), L_1(X,Y),M_1(X,Y)\in\mathbb{Z}[X,Y]$ with
$$L_0D+M_0E=R\cdot X^{2n-1}$$ and $$L_1D+M_1E=R\cdot Y^{2n-1}$$
I encountered this problem while preparing a talk about the proof of the Mordell-Weil theorem for elliptic curves over $\mathbb{Q}$ and don't know how to solve it.
If $D(x)$ and $E(x)$ are polynomials of degree $n$ with coefficients in a commutative ring, then it is a standard property of resultants that there are polynomials $L$ and $M$ of degree $n-1$ such that $$L(x)D(x)+M(x)E(x)=R$$ where $R$ is the resultant of $D$ and $E$.
Given forms $D(x,y)$ and $E(x,y)$, we get $$L(x)D(x,1)+M(x)E(x,1)=R$$ then we let $x=X/Y$ and multiply by $Y^{2n-1}$ to get $$L_1(X,Y)D(X,Y)+M_1(X,Y)E(X,Y)=RY^{2n-1}$$ and we do the same with $D(1,y)$ and $E(1,y)$ to get the other equation you want.