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Explanation:

If p² + q² - 29 = 2pq - 20 = 52 - 2pq, then the difference between the maximum and minimum possible value of (p³ - q³) is

Given, p² + q² - 29 = 2pq - 20
⇒ p² + q² - 2pq = 29 - 20
⇒ (p - q)² = 9
⇒ p - q = ± 3

Also given, p² + q² - 29 = 52 - 2pq
⇒ p² + q² + 2pq = 81
⇒ (p + q)² = 81
⇒ p + q = ± 9

Case 1: p - q = + 3 and p + q = + 9
Solving these 2 equations we get, p = 6 and q = 3
∴ p³ - q³ = 216 - 27 = 189

Case 2: p - q = - 3 and p + q = + 9
Solving these 2 equations we get, p = 3 and q = 6
∴ p³ - q³ = 27 - 216 = - 189

Case 3: p - q = + 3 and p + q = - 9
Solving these 2 equations we get, p = - 3 and q = - 6
∴ p³ - q³ = (-27) - (-216) = 189

Case 4: p - q = - 3 and p + q = - 9
Solving these 2 equations we get, p = - 6 and q =  - 3
∴ p³ - q³ = - 216 - (-27) = - 189

∴ Highest possible value of p³ - q³ = 189 least possible value of p³ - q³ = - 189.

∴ Required difference = 189 - (-189) = 378

Hence, option (b).