Factorize each of the following:

(i) Using identity,

(a + b)³ = a³ + b³ + 3a²b + 3ab²

8a³ + b³ + 12a²b + 6ab²

= (2a)³ + b³ + 3 (2a) (2b) + 3 (2a) (b)²

= (2a + b)³

= (2a + b) (2a + b) (2a + b)

(ii) Using identity,

(a + b)³ = a³ + b³ + 3a²b + 3ab²

8a³ – b³ – 12a²b + 6ab²

= (2a)³ – b³ – 3 (2a)²b + 3 (2a) (b)²

= (2a – b)³

= (2a – b) (2a – b) (2a – b)

(iii) Using identity,

(a + b)³ = a³ + b³ + 3a²b + 3ab²

27 – 125a³ – 135a + 225a²

= 3³ – (5a)³ – 3 (3)²(5a) + 3 (3) (5a)²

= (3 – 5a)³

= (3 – 5a) (3 – 5a) (3 – 5a)

(iv) Using identity,

(a + b)³ = a³ + b³ + 3a²b + 3ab²

64a³ – 27b³ – 144a²b + 108ab²

= (4a)³ – (3b)³ – 3 (4a)² (3b) + 3 (4a) (3b)²

= (4a – 3b)²

= (4a – 3b) (4a – 3b) (4a – 3b)

(v) Using identity,

(a + b)³ = a³ + b³ + 3a²b + 3ab²

27p³ - 1/216 - 9/2p² + 1/4p

= (3p)³ – (1/6)³ – 3 (3p)² (1/6) + 3 (3p) (1/6)²

= (3p - 1/6)³

= (3p - 1/6) (3p - 1/6 (3p - 1/6)