Divide through by y^3: (x/y)^3+3(x/y)^2-4-x/y^3+1/y^2+3/y^3=0.
When y is very large, the last 3 terms become small, so (x/y)^3+3(x/y)^2-4=0 approximately, and (x/y)=1, therefore y=x is an asymptote. Similarly, dividing through by x^3 gives 1+3(y/x)-4(y/x)^3=0 approx. y=x is a solution to this equation.
If we write z=y/x, then 1+3z-4z^3=0, and z=1 is a root.
Synthetic division:
1 | -4 0 3 1
-4 -4 -4 -1
-4 -4 -1 | 0 = -4z^2-4z-1 = -(2z+1)^2 is the other factor: 1+3z-4z^3=(1-z)(2z+1)^2=0. From this, z=-1/2 is the other root, therefore y/x=-1/2 and y=-x/2 is the other asymptote.
SOLUTION: the asymptotes are y=x and y=-x/2.