Base Case: For \(n = 1\), \(6^4 + 38^1 - 2 = 1332\), which is divisible by 74.

Inductive Step: Assume \(6^{4k} + 38^k - 2\) is divisible by 74 for some positive integer \(k\).

Then, consider \(6^{4(k+1)} + 38^{k+1} - 2\):

\(= (1295 + 1) \times 6^{4k} + (37 + 1) \times 38^k - 2\)

This expression is divisible by 74 because \(1295 \times 6^{4k} + 37 \times 38^k\) is divisible by 74.

Hence, by induction, \(6^{4n} + 38^n - 2\) is divisible by 74 for every positive integer \(n\).