Let a polynomial of $2n$-variables be $$ f(x_1,\cdots,x_n,y_1,\cdots,y_n)=\prod_{i,j=1}^n(1+x_i+y_j). $$ Let the elementary symmetric polynomials be $\alpha_1=\sum_{i=1}^n x_i$, $\alpha_2=\sum_{i<j} x_ix_j$, $\cdots$, $\alpha_n=\prod_{i=1}^n x_i$; $\beta_1=\sum_{i=1}^n y_i$, $\beta_2=\sum_{i<j} y_iy_j$, $\cdots$, $\beta_n=\prod_{i=1}^n y_i$.
I want to express $f$ in terms of $\alpha_1,\cdots,\alpha_n$ and $\beta_1,\cdots,\beta_n$. What is the final explicit expression? I lost in the complicated computations.
Your product can be rewritten as $$ Res(f(X),Res_Y(g(Y),1+X+Y))=Res(f(X), g(-1-X)) $$ where $f$ is the polynomial with roots $x_j$ and $g$ is the polynomial with roots $y_j$.
Since the coefficients of the polynomials are signed variants of the elementary symmetric polynomials, this gives a -- not very explicit -- polynomial expression in the elementary symmetric polynomials of the collections $(x_j)$ and $(y_j)$