What is the eigenvector(s) of this matrix?

First you have to solve the eigen vaiue equation'
(-2-l, 1) (-2, -4-l) =(2+l) (4+l) +2=0
8+4 l +2l +l^2+2=0
l^2 +6 l +10=0.
We l= (-6 +/- sqrt(36-40))/2
l= (-3 +/- i )
Let the eigen values be
l1 = -3 + i, and l2= -3 -i.
Now to find the eigen vector.
You yse the derminant. You get
(-2-l1) x +y=0.
Select y=1,
This gives
x= 1/(l1+2). The eigen vectors are ( 1/(l1+2), 1)
Similarly the other eigen vector.