I haven't been through the arithmetic, but I can offer a few comments.

Complex eigenvalues occur as conjugate pairs, in which case there can't be a single real eigenvalue, there have to be none, two or four.

If the original matrix contains only real elements and an eigenvalue is real, then the associated eigenvector will contain real components only.

Check your algebra, I think that the 's should be 's, and I think that the negative sign at the front of the first component shouldn't be there.