To solve the given expression, we can expand both terms and simplify.

Expanding the first term:
(5x + y - 6)^2 = (5x + y - 6)(5x + y - 6)
= 25x^2 + 5xy - 30x + 5xy + y^2 - 6y - 30x - 6y + 36
= 25x^2 + 10xy - 60x + y^2 - 12y + 36

Expanding the second term:
(3x - y - 2)^4 = (3x - y - 2)(3x - y - 2)(3x - y - 2)(3x - y - 2)
= (3x - y - 2)^2 (3x - y - 2)^2
= ((3x - y - 2)(3x - y - 2))^2
= (9x^2 - 3xy - 6x - 3xy + y^2 + 6y - 6x + 2y + 4)^2
= (9x^2 - 6xy - 12x + y^2 + 8y + 4)^2

Adding both expressions:
(25x^2 + 10xy - 60x + y^2 - 12y + 36) + (9x^2 - 6xy - 12x + y^2 + 8y + 4)^2 = 0

Simplifying the combined expression further may not be practical, so the expression as it stands is the solution.