A Followup on Solving A Fourth Degree Equation

Acknowledgment

I learned of a problem posed by Dan Sitaru from a solution by Kunihiko Chikaya. More than the solution I liked the question, and not even the question but its being a followup on the previous one. This teaches a brilliant way to generate new problems by modifying the ones already solved. This is certainly an excellent illustration of George Polya's last step - Looking back - in problem solving.

Problem

Find $\displaystyle\sum_{i=1}^{4}|x_i|,\;$ where $x_i,\;$ $i=1,2,3,4\;$ are the roots of

$x^4+8x^3+23x^2+28x+10=0$

Solution

Clearly this problem is a followup on another one where a similar equation has been solved by three different methods. Any of these will be a good first step for answering the question at hand. I'll use the second solution which implies that

$x^4+8x^3+23x^2+28x+10=(x+2)^4-(x+2)^2-2.$

Thus we are led to four roots of the given polynomial: $-2\pm\sqrt{2}\;$ and $2\pm i,\;$ whose moduli add up to $4+2\sqrt{5}.$

71867219