For some n>4, can one find two symmetric polynomials $S_1$ and $S_2$ in $\mathbb{Q}[x_1,...,x_n]$ such that $S_1+x_1S_2$ is a square in $\mathbb{Q}[x_1,...,x_n]$?

I have such a construction for the case $n \leq 4$, and my construction for n=4 itself is already quite tricky. I suspect there is no such construction for $n>4$, but I'm still struggling to find the best theoretical approach. Any suggestion?