Question

Simplify the following: 

(a + 2b) (a² – 2ab + 4b²) + (a – 2b) (a² + 2ab + 4b²)

Answer 1

Given: (a + 2b)(a² – 2ab + 4b²) + (a – 2b)(a² + 2ab + 4b²)

Step-wise calculation:

1. Recognize the identities: 

(a + 2b)(a² – 2ab + 4b²) = a³ + (2b)³

(a + 2b)(a² – 2ab + 4b²) = a³ + 8b³ 

And (a – 2b)(a² + 2ab + 4b²) = a³ – (2b)³

(a – 2b)(a² + 2ab + 4b²) = a³ – 8b³

2. Substitute these into the expression:

a³ + 8b³ + a³ – 8b³

3. Simplify by combining like terms:

a³ + a³ + 8b³ – 8b³ = 2a³ + 0

a³ + a³ + 8b³ – 8b³ = 2a³