You have x1'= -x1- 4x2 and x2'= -x1- x2. From the first equation, 4x2= -x1'- x1. If you differentiate the first equation again, you get x1"= -x1'- 4x2'. From the second equation, x2'= -x1- x2 so that is x1"= -x1'- 4(-x1- x2)= -x1'- 4x1+ 4x2= -x1'- 4x1+ (-x1'- x1)= -2x1'- 5x1.
Can you solve the single equation x1"= -2x1'- 5x1 which is the same as x1"+ 2x1'+ 5x1= 0? If so then you can find x2 from 4x2= -x1'- 1.
Another way to do this problem, more in the "matrix" spirit, would be to find the eigenvalues and eigenvectors of