2021 AMC 12A Problem 21

The five solutions to the equation(z-1)(z^2+2z+4)(z^2+4z+6)=0 may be written in the form x_k+y_ki for 1\le k\le 5, where x_k and y_k are real. Let \mathcal E be the unique ellipse that passes through the points (x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4), and (x_5,y_5). The eccentricity of \mathcal E can be written in the form \sqrt{\frac mn}, where m and n are relatively prime positive integers. What is m+n? (Recall that the eccentricity of an ellipse \mathcal E is the ratio \frac ca, where 2a is the length of the major axis of \mathcal E and 2c is the is the distance between its two foci.)

\textbf{(A) }7 \qquad \textbf{(B) }9 \qquad \textbf{(C) }11 \qquad \textbf{(D) }13\qquad \textbf{(E) }15

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.