(i) (2x + 3y + 4z)² 

(a + b + c)² ≡ a² + b² + c² + 2ab + 2bc + 2ca 

∴ (2x + 3y + 4z)² = (2x)² + (3y)² + (4z)² + 2(2x) 

(3y) + 2(3y) (4z) + 2(4z) (2x) 

= 4x² + 9y² + 16z² + 12xy + 24yz + 16zx

(ii) (-p + 2q + 3r)² 

(a + b + c)² ≡ a² + b² + c² + 2ab + 2bc + 2ca 

(-p + 2q + 3r)² = (-p)² + (2q)² + (3r)² + 2(-p) (2q) + 2(2q) (3r) + 2(-p) (3r) 

= p² + 4q² + 9r² – 4pq +12qr – 6rp

(iii) (2p + 3)(2p – 4)(2p – 5) 

(x + a) (x + b) (x + c) ≡ x³ + (a + b + c)x² + (ab + be + ca) x + abc 

(2p + 3)(2p – 4)(2p – 5) = (2p)³ + (3 – 4 – 5) (2p)² + [(3 x – 4)² + (-4 x -5) + (-5 x 3)] 2p + 3 x – 4 x – 5 

= 8p³ + (-6) (4p²) + [-12 + 20 + (-15)] 2p + 60 

= 8p³ – 24p² + (-7)2p + 60 

(2p + 3)(2p – 4)(2p – 5) = 8p³ – 24p² – 14p + 60

(iv) (3a + 1)(3a – 2)(3a + 4) 

(x + a) (x + b) (x + c) ≡ x³ + (a + b + c)x² + (ab + bc + ca)x + abc 

(3a + 1)(3a – 2)(3a + 4) = (3a)³ + (1 – 2 + 4) (3a)² + [1 x (- 2) + (-2 x 4) + 4 x 1] (3a) + 1 x -2 x 4 

= 27a³ + 3 (9a²) + (-2 – 8 + 4)3a – 8 

= 27a³ + 27a² – 8a – 8