Given two polynomials and , let a, b, c, d be the roots of , find

First, notice that

We know that plugging a, b, c, or d into gives 0, so we have to find the sum of squares of the roots minus the sum of the roots plus 4.

By Vieta’s Theorem, the sum of the roots equals the negative ratio of the coefficients of degree 3 and 4, that is to say 1.

Also by Vieta’s Theorem, the sum of the products of any two roots equals the ratio of the coefficients coefficients of degree 2 and 4, that is to say -1.

So the sum of the squares equals

So the final sum equals