`a)`
`(x-2)^3 - (x+2)^3`
` = (x^3 - 3 . x^2 . 2 + 3 . x . 2^2 - 2^3) - (x^3 + 3 . x^2 . 2 + 3 . x . 2^2 + 2^3) `
` = (x^3 - 6x^2 + 12x - 8) - (x^3 + 6x^2 + 12x + 8) `
`= x^3 - 6x^2 + 12x - 8 - x^3 - 6x^2 - 12x - 8`
` = (x^3 - x^3) - (6x^2 + 6x^2) + (12x - 12x) - (8+8)`
` = -12x^2 - 16`
`b)`
`(x-1)(x^2 + x +1) + (x+1)(x^2 - x + 1)`
`= (x-1)(x^2 + x . 1 + 1^2) + (x+1)(x^2 - x . 1 + 1^2)`
` = (x^3 - 1^3) + (x^3 + 1^3)`
` = (x^3 - 1) +(x^3+1)`
`= x^3 - 1 + x^3 + 1`
` = (x^3 + x^3) + (1-1)`
` = 2x^3`
`d)`
`x(x-2)(x+2) - (x-2)^3`
`= x (x^2 - 2^2) - (x^3 - 3 . x^2 . 2 + 3 . x . 2^2 - 2^3)`
`= x (x^2 - 4) - (x^3 - 6x^2 + 12x - 8) `
` = x^3 - 4x - x^3 + 6x^2 - 12x + 8`
` = (x^3 - x^3) + 6x^2 - (4x+12x) + 8`
`= 6x^2 - 16x + 8`
`e)`
`(2x+1)(4x^2 -2x+1) + (2x-1)(4x^2 + 2x+1)`
`= (2x+1)[ (2x)^2 - 2x . 1 + 1^2] + (2x-1) [ (2x)^2 + 2x . 1 + 1^2]`
` = [ (2x)^3 + 1^3] + [ (2x)^3 - 1^3]`
`= (8x^3 + 1) + (8x^3 - 1)`
`= 8x^3 + 1 + 8x^3 -1`
` = (8x^3 + 8x^3) + (1-1)`
`= 16x^3`
`f)`
`(2+3x)(4 - 6x + 9x^2) - (2-3x)(4 + 6x + 9x^2)`
`= (2+3x) [ 2^2 - 2 . 3x + (3x)^2] - (2-3x) . [ 2^2 + 2 . 3x + (3x)^2]`
`= [ 2^3 + (3x)^3] - [ 2^3 - (3x)^3]`
` = (8 + 27x^3) - (8 - 27x^3)`
`= 8 + 27x^3 - 8 + 27x^3`
` = (27x^3 + 27x^3) + (8-8)`
` = 54 x^3`
`g)`
`(xy^2 - 3)^2 - (y-2x)^3 - x^2y (y^3 -12) - 8x^3 + y^3`
` = [ (xy)^2 - 2 . xy^2 . 3 + 3^2] - [y^3 - 3 . y^2 . 2x + 3 . y . (2x)^2 - (2x)^3] - (x^2y^4 -12x^2y) - 8x^3 + y^3`
` = (x^2y^4 - 6xy^2 + 9) - (y^3 - 6xy^2 + 12x^2y - 8x^3) - x^2y^4 + 12x^2y - 8x^3 + y^3`
` = x^2y^4 - 6xy^2 + 9 - y^3 + 6xy^2 - 12x^2y + 8x^3 - x^2y^4 + 12x^2y - 8x^3 + y^3`
` = (x^2 y^4 - x^2 y^4) + (6xy^2 - 6xy^2) + (y^3 - y^3) + (8x^3 - 8x^3) + (12x^2y -12x^2y) + 9`
` = 9`
