1.6.15 Fields, Groups, Rings

Problem: Express the symmetric polynomial \[ \left(x_{1} x_{2}+x_{3} x_{4}\right)\left(x_{1} x_{3}+x_{2} x_{4}\right)\left(x_{1} x_{4}+x_{2} x_{3}\right) \] in the ring \( \mathbb{Q}\left[x_{1}, x_{2}, x_{3}, x_{4}\right] \) in terms of elementary symmetric functions.