Find the group of permutations on $\{1, 2, 3, 4\}$ which leaves the symmetric polynomial $x_1 x_2+x_3x_4$ invariant.
What I know about this is as follows:
A polynomial $f(x_1, . . . , x_n)$ is invariant under $S_n$ if for all $\pi \in S_n$ $$f(\pi(x_1), . . . , \pi(x_n)) = f(x_1, . . . , x_n)$$ But here how will I find the permutation such that the polynomial is invariant.
Well, if $\sigma\in G$, the group you want to find, then you have: $$\sigma(x_1)\sigma(x_2)+\sigma(x_3)\sigma(x_4) = x_1x_2+x_3x_4$$.
The key here is that the above identity should be thought of as an equality between two polynomials. Can you take it from here?