Step-by-step Solution:

Pair A: 8:81::64: ?
- Given Relationship: \(x^1 : (x + 1)^{(x + 1)}\).
- Calculation:
- \(8 = 2^3\), so \(x = 2\).
- The next term is \((2 + 1)^{(2 + 1)} = 3^3 = 27\).
- However, the given pair is \(8:81\), which suggests \(x = 8\) and \((8 + 1)^{(8 + 1)} = 9^9\), which is not practical.
- Alternatively, if we consider \(8 = 2^3\) and \(81 = 3^4\), the relationship is \(2^3 : 3^4\).
- For \(64 = 4^3\), the next term would be \(5^4 = 625\).
- Match: A-IV (625).



Pair B: 182: ? ::210: 380
- Given Relationship: \(x = n^2 - n\).
- Calculation:
- \(210 = 15^2 - 15 = 225 - 15\).
- \(380 = 20^2 - 20 = 400 - 20\).
- The difference between \(15\) and \(20\) is \(5\).
- Similarly, \(182 = 14^2 - 14 = 196 - 14\).
- The next term would be \((14 + 5)^2 - (14 + 5) = 19^2 - 19 = 361 - 19 = 342\).
- Match: B-III (342).



Pair C: 42:56:110: ?
- Given Relationship: \(x = n^2 - n\).
- Calculation:
- \(42 = 7^2 - 7 = 49 - 7\).
- \(56 = 8^2 - 8 = 64 - 8\).
- \(110 = 11^2 - 11 = 121 - 11\).
- The next term would be \(12^2 - 12 = 144 - 12 = 132\).
- Match: C-II (132).



Pair D: 48:122::168: ?
- Given Relationship: \(x = n^2 \pm 1\).
- Calculation:
- \(48 = 7^2 - 1 = 49 - 1\).
- \(122 = 11^2 + 1 = 121 + 1\).
- \(168 = 13^2 - 1 = 169 - 1\).
- The next term would be \(17^2 + 1 = 289 + 1 = 290\).
- Match: D-I (290).



Final Matching:
- A-IV: 8:81::64:625.
- B-III: 182:342::210:380.
- C-II: 42:56:110:132.
- D-I: 48:122::168:290.



Answer:
\[
\text{A-IV, B-III, C-II, D-I}
\]