This might sound like a noob method but i moved the 4 to the left side to get
x^2+x+1/x+1/x^2-4=0
then i did the same thing as flying mullet and made one denominator
(x4 + x3 + x + 1 - 4)/x2 = 0

So i just ignored that bottom x2 because i can move it to the other side
Then i tested values for x4+x3+x+1-4 until it = 0 to find a factor
1 was a factor
(x-1)(x3+2x2-2x-1)=0
Then i subed for x3+2x2-2x-1 until i found a factor
1 was a factor
(x-1)(x-1)(x2+3x+1)
from here u get x=1 or x2+3x+1=0 which u can use the quadratic formula for x = (-3+root5)/2

Ive never been good at remembering or applying any of the larger theorems or rule so this might not be the quickest way