what happens when the eigenvalues consist of 2 real roots and 1 complex conjugate pair?
This means that the one complex conjugate root is still an oscillation but the other has split into two aperiodic movements. One of them should be a positive and the other a negative real value. While the negative root signifies a damped, aperiodic movement, the positive root shows that the airplane has one instable, divergent mode.
Can this happen?
Yes, for example when the center of gravity is moved aft of the neutral point, the phygoid will split into two real roots. The positive one describes the divergent behavior when static stability has been lost.
The equations of motion describe how acceleration, speed and pitch rate of the airplane react to small changes of the control parameters (elevator angle and throttle setting). They are second order differential equations and can be solved similar to systems of quadratic equations where the coefficients form the elements of a square matrix. The eigenvalues of the matrix describe frequency and damping of the movement which is caused by changes to the control parameters while the eigenvectors describe how acceleration, speed and pitch rate are affected in detail.
Note that this only concerns longitudinal movement (forward-back, up-down and pitch). The equations can be expanded to cover roll, yaw and left-right movement as well, but the question only concerns the simpler longitudinal part.