Let us simplify u

_{i. (a + b)(a – b)(a}² + ab + b²)(a² – ab + b²)

= (a + b)(a² – ab + b²)(a – b)(a² + ab + b²)

Using x³ – y³ = (x – y)(x² + xy + y²)

x³ + y³ = (x + y)(x² – xy + y²)

= (a³ + b³)(a³ – b³)

Now, using (a + b)(a – b) = a² – b²

= a⁶ – b⁶

ii. (a – 2b)(a² + 2ab + 4b²)(a³ + 8b³)

= (a – 2b)(a² + a(2b) + (2b)²)(a³ + (2b)³)

Using x³ – y³ = (x – y)(x² + xy + y²)

x³ + y³ = (x + y)(x² – xy + y²)

= (a³ – (2b)³)(a³ + (2b)³)

Now, using (a + b)(a – b) = a² – b²

= a⁶ – (2b)⁶

= a⁶ – 64b⁶

iii. (4a² – 9)(4a² – 6a + 9)(4a² + 6a + 9)

= [(2a)² – 3²](4a² – 6a + 9)(4a² + 6a + 9)

Using x² – y² = (x – y)(x + y)

= (2a – 3)(2a + 3)(4a² – 6a + 9)(4a² + 6a + 9)

= (2a – 3)(4a² + 6a + 9)(2a + 3)(4a² – 6a + 9)

Using x³ – y³ = (x – y)(x² + xy + y²)

x³ + y³ = (x + y)(x² – xy + y²)

= [(2a)³ – 3³][(2a)³ + 3³]

= (2a)⁶ - 3⁶

= 64a⁶ – 729

iv. (x – y)(x² + xy + y²) + (y – z)(y² + yz + z²) + (z – x)(z² + zx + x²)

Using a³ – b³ = (a – b)(a² + ab + b²)

= x³ – y³ + y³ – z³ + z³ – x³

= 0

v. (x + 1)(x² - x + 1) + (2x – 1)(4x² + 2x + 1) – (x – 1)(x² + x + 1)

= (x + 1)(x² - x + 1) + (2x – 1)((2x)² + 2x + 1) – (x – 1)(x² + x + 1)

Using x³ – y³ = (x – y)(x² + xy + y²)

x³ + y³ = (x + y)(x² – xy + y²)

= x³ + 1 + (2x)³ – 1 - (x³ – 1)

= x³ + 1 + 8x³ – 1 – x³ + 1

= 8x³ + 1