Expanding each term in the expression:

= (3a^2 + 4)^2 + (3a^2 - 4)^2 - 2(5 - 3a^2)(5 + 3a^2) - 6(6a^4 + 1)
= (3a^2 + 4)(3a^2 + 4) + (3a^2 - 4)(3a^2 - 4) - 2(25 - 9a^4) - 6(6a^4 + 1)
= (9a^4 + 12a^2 + 12a^2 + 16) + (9a^4 - 12a^2 - 12a^2 + 16) - 50 + 18a^4 - 36a^4 - 6
= 18a^4 + 24a^2 + 24a^2 + 32 + 18a^4 - 24a^2 - 24a^2 + 32 - 50 + 18a^4 - 36a^4 - 6
= 18a^4 + 18a^4 + 18a^4 + 24a^2 - 24a^2 + 18a^4 - 36a^4 + 32 + 32 - 50 - 6
= 72a^4 + 32 + 32 - 50 - 6
= 72a^4 + 8

Therefore, the simplified form of the expression is 72a^4 + 8.