`24x^2 - 3x`

  `= 3x (8x^3 - 1)`

  `= 3x{(2x)^3 - 1}`

  `= 3x(2x - 1) {(2x)^2 + 2x.1 + (1)^2}`

  `= 3x(2x - 1) (4x^2 + 2x + 1)`  (Ans)

`(a^2 + b^2)^3 + 8a^3b^3`

  ধরি, `a^2 + b^2 = p`

 :. প্রদত্ত রাশি

    `= p^3 + 8a^3b^3`

    `= p^3 + (2ab)^3`

    `= (p + 2ab) {p^2 - p.2ab + (2ab)^2}`

    `= (p + 2ab) (p^2 - 2abp + 4a^2b^2)`

    `= (a^2 + b^2 + 2ab) {(a^2 + b^2)^2 - (a^2 + b^2)2ab + 4a^2b^2}`

                           [p এর মান বসিয়ে]

     `= (a + b)^2 {a^4 + b^4 + 2a^2b^2 - 2a^3b - 2ab^3 + 4a^2b^2}`

     `= (a + b)^2 (a^4 - 2a^3b + 6a^2b^2 - 2ab^3 + b^4)`   (Ans)

`x^3 + 3x^2 + 3x + 2`

  `= x^3 + 3x^2 + 3x + 1 + 1`

  `= (x)^3 + 3.x^2.1 + 3.x.1^2 + (1)^3 + 1`

  `= (x + 1)^3 + (1)^3`

  `= {(x + 1) + 1} {(x + 1)^2 - (x + 1) 1 + (1)^2}`

  `= (x + 1 + 1) (x^2 + 2x + 1 - x - 1 + 1)`

  `= (x + 2) (x^2 + x + 1)`  (Ans)

`a^3 - 6a^2 + 12a - 9`

  `= a^3 - 6a^2 + 12a - 8 - 1`

  `= a^3 - 3.a^2.2 + 3.a.2^2 - 2^3 - 1`

  `= (a - 2)^3 - 1`

  `= (a - 2)^3 - (1)^3`

  `= (a - 2 - 1) {(a - 2)^2 + (a - 2).1 + 1^2}`

  `= (a - 3) (a^2 - 4a + 4 + a - 2 + 1)`

  `= (a - 3) (a^2 - 3a + 3)`   (Ans)

`a^3 - 9b^3 + (a + b)^3`

  `= a^3 - b^3 + (a + b)^3 - 8b^3`

  `= (a - b) (a^2 + ab + b^2) + (a + b)^3 - (2b)^3`

  `= (a - b) (a^2 + ab + b^2) + (a + b - 2b) {(a + b)^2 + (a + b).2b + (2b)^2}`

  `= (a - b) (a^2 + ab + b^2) + (a - b) (a^2 + 2ab + b^2 + 2ab + 2b^2 + 4b^2)`

  `= (a - b) (a^2 + ab + b^2) + (a - b) (a^2 + 4ab + 7b^2)`

  `= (a - b) (a^2 + ab + b^2 + a^2 + 4ab + 7b^2)`

  `= (a - b) (2a^2 + 5ab + 8b^2)`  (Ans)