Question 121819
Let z=x+1/x then z^2+4z+3=0 then z=-1 or -3 then

x+1/x =-1 or x+1/x=-3  

if x+1/x = -1 then x^2+1=-x then x^2+x+1=0 ( no real solution)
if x+1/x = -3 then x^2+1=-3x then x^2+3x+1=0 then

x=(-3+sqrt(5))/2 or

x=(-3-sqrt(5))/2