`4x^4 - 27x^2 - 81`

  `= 4x^4 - 36x^2 + 9x^2 - 81`

  `= 4x^2(x^2 - 9) + 9(x^2 - 9)`

  `= (x^2 - 9) (4x^2 + 9)`

  `= (x^2 - 3^2) (4x^2 + 9)`

   `= (x + 3) (x - 3) (4x^2 + 9)`  (Ans)

`ax^2 + (a^2 + 1) x + a`

  `= ax^2 + a^2x + x + a`

  `= ax(x + a) + 1(x + a)`

  `= (x + a) (ax + 1)`  (Ans)

`3(a^2 + 2a)^2 - 22(a^2 + 2a) + 40`

  `= a^2 + 2a = p`

  `= 3p^2 - 22p + 40`

  `= 3p^2 - 12p - 10p + 40`

  `= 3p(p - 4) - 10(p - 4)`

  `= (p - 4) (3p - 10)`

  `= (a^2 + 2a - 4) {3(a^2 + 2a) - 10}`  [p]

  `= (a^2 + 2a - 4) (3a^2 + 6a - 10)`    (Ans)

`14(x + z)^2 - 29(x + z) (x + 1) - 15(x + 1)^2`

   ধরি, `x + z = p`

   এবং `x + 1 = q`

  প্রদত্ত রাশি, 
     `= 14p^2 - 29pq - 15q^2`

     `= 14p^2 - 35pq + 6pq - 15q^2`

     `= 7p(2p - 5q) + 3q(2p - 5q)`

     `= (2p - 5q) (7p + 3q)`

     `= {2(x + z) - 5(x + 1)} {7(x + z) + 3(x + 1)}`

                                  [p, q এর মান বসিয়ে]

      `= (2x + 2z - 5x - 5) (7x + 7z + 3x + 3)`

      `= (2z - 3x - 5) (10x + 7z + 3)`  (Ans)

`(a - 1)x^2 + a^2 xy + (a + 1) y^2`

    `a - 1 = p`

     `a + 1 = q`
   ------------------------------
     :.` (a - 1) (a + 1) = pq`   [গুণ করে]

         `a^2 - 1 = pq`

        `:. a^2 = pq + 1`

  প্রদত্ত রাশি  

 `= px^2 + (pq + 1) xy + qy^2`

 `= px^2 + pqxy + xy + qy^2`
 
`= px(x + qy) + y(x + qy)`

`= (x + qy) (px + y)`

`= {x + (a + 1)y} {(a - 1) x + y} `[p q এর মান বসিয়ে]

`= (x + ay + y) (ax - x + y)`   (Ans)