Polynomial roots — Quartic with all terms

Question

For with roots , find .

This practice question was inspired by Question 11(h) in the 2021 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: translate the requested symmetric expression into sums and products of roots from the coefficients., product .
State the final answer clearly in the form the question requested..

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: Answer .

Any mark labels are a TestMum study aid, not an official NESA marking allocation.