\( ( a ^ { n } - 1 ) ^ { 2 } ( a ^ { n } + 1 ) ^ { 2 } - a ^ { n } ( a ^ { n } - 1 ) ^ { 3 } - a ^ { 2 n } ( 3 a ^ { n } - 5 ) \)

To simplify the expression \( ( a ^ { n } - 1 ) ^ { 2 } ( a ^ { n } + 1 ) ^ { 2 } - a ^ { n } ( a ^ { n } - 1 ) ^ { 3 } - a ^ { 2 n } ( 3 a ^ { n } - 5 ) \), it can be helpful to factor out common terms or to regroup the terms strategically. For instance, recognizing that \( (a^n - 1)^2 \) is common in the first two parts of the expression allows us to set up a clearer path to cancellation or reduction when we collect like terms. Additionally, substitution methods can also facilitate a more straightforward simplification. Let \( x = a^n \); it transforms the expression into a polynomial in terms of \( x \). This can make it easier to apply polynomial identities or recognize patterns for factoring. After performing these steps, you may find that the expression can be reduced greatly, providing a concise final form.