Let's simplify the given expression step by step.

Start by squaring the expression within the parentheses:
(3y^-1 - x^-2)^2 = (3y^-1 - x^-2)(3y^-1 - x^-2)
Expanding this using the distributive property:
= 9y^-2 - 3y^-1x^-2 - 3x^-2y^-1 + x^-4

Now plug this back into the original expression:
6x^-2y^-1 + (9y^-2 - 3y^-1x^-2 - 3x^-2y^-1 + x^-4)

Simplify by combining like terms:
6x^-2y^-1 + 9y^-2 - 3y^-1x^-2 - 3x^-2y^-1 + x^-4
= 9y^-2 + 6x^-2y^-1 - 3y^-1x^-2 - 3x^-2y^-1 + x^-4

Therefore, the simplified expression is 9y^-2 + 6x^-2y^-1 - 3y^-1x^-2 - 3x^-2y^-1 + x^-4.