Simplify:

(i) (2x + y + 4z)(4x² + y² + 16z² – 2xy – 4yz – 8zx)

(ii) (x – 3y – 5z)(x² + 9y² + 25z² + 3xy – 15yz + 5zx)

(i) (2x + y + 4z) (4x² + y² + 16z² – 2xy – 4yz – 8zx)

⇒ Here the identity used is

a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)

Comparing this with given condition

(2x + y + 4z) (4x² + y² + 16z² – 2xy – 4yz – 8zx) = (2x)³ + y³ + (4z)³ – 3 × 2x × y × 4z

⇒ 8x³ + y³ + 64z³ – 24xyz

(ii) (x – 3y – 5z)(x² + 9y² + 25z² + 3xy – 15yx + 5zx)

⇒ Here the identity used is

a³ + b³ + c³ – 3abc = (a + b + c) (a² + b² + c² – ab – bc – ac)

Comparing this with given condition

(x – 3y – 5z)(x² + 9y² + 25z² + 3xy – 15yx + 5zx)

= x³ + (– 3y)³ + (– 5z)³ – 3(x) (– 3y)(– 5z)

= x³ – 27y³ – 125z³ – 45xyz