Factorize:
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For quadratic polynomial. We factorize the middle term such that the product of the terms is equal to the product of the last term and coefficient of x² and sum or difference is equal to the middle term.

(i)

12x² - 7x + 1

For the equation given above:

7 is to be factorized such that, the product of two terms is 12 x 1 = 12 and the sum is equal to 7
This can be done by 3 and 4
as 3 x 4 = 12
and 3 + 4 = 7
Therefore,

12x² - 7x + 1 = 12x² – 4x – 3x + 1

 = 4x (3x – 1) – 1 (3x – 1)

 = (3x – 1) (4x – 1)

(ii) 

2x² + 7x + 3

For the equation given above:

7 is to be factorized such that, the product of two terms is 3 x 2 = 6 and the sum is equal to 7
This can be done by 6 and 1
as 6 x 1 = 6
and 6 + 1 = 7
Therefore,

2x² + 7x + 3 = 2x² + 6x + x + 3

= 2x (x + 3) + 1 (x + 3)

 = (x + 3) (2x + 1)

(iii)

6x² + 5x – 6

For the equation given above:

7 is to be factorized such that, the product of two terms is 6 x 6 = 36 and the difference is equal to 5
This can be done by 9 and 4
as 9 x 4 = 36
and 9 - 4 = 5
Therefore,

6x² + 5x – 6 = 6x² + 9x – 4x – 6

 = 3x (2x + 3) – 2 (2x + 3)

 = (2x + 3) (3x – 2)

(iv)

3x² - x - 4

For the equation given above:

7 is to be factorized such that, the product of two terms is 4 x 3 = 12 and the difference = 1
This can be done by 4 and 3
as 4 x 3 = 12
and 4 - 3 = 1
Therefore,

3x² - x - 4=3x² – 4x + 3x – 4

= x (3x – 4) + 1 (3x – 4)

= (3x – 4) (x + 1)