To simplify this expression, we will first expand and simplify the square of (6√6 - √13):

(6√6 - √13)²
= (6√6 - √13)(6√6 - √13)
= (6√6)(6√6) - (6√6)(√13) - (√13)(6√6) + (√13)(√13)
= 36(6) - 6√(613) - 6√(613) + 13
= 216 - 6√78 - 6√78 + 13
= 229 - 12√78

Now, we simplify the expression:

(6√6 - √13)² + 12(√78 + 4)
= (229 - 12√78) + 12√78 + 48
= 277

Therefore, (6√(6)-√(13))² + 12(√(78)+4) simplifies to 277.