This method involves splitting the middle term into two parts which can then be paired with the numbers either side and factorised, so that both bracket contains exactly the same terms.
This bracket then becomes one term, the other is found by collecting what is left and putting it into a bracket.
This is really just a reversal of FOIL.
To find the correct split, the numbers must add together to make the middle number (coefficient of x) and multiply together to make the first number times the last number (coefficient of x2 times the constant).
Example
Factorise x2 + 4x + 3
The middle term is 4x
Two numbers are needed which add to give 4
And which multiply to give 3
Use 1 and 3, so 4x can be split into 3x + x
x2 + 4x + 3
= x2 + x + 3x + 3
factorise both parts
= x(x + 1) + 3(x + 1) both brackets are same
=(x+3)(x+1)
Example
Factorise x2 - 7x – 8
The middle term is -7x
Two numbers are needed which add to give -7
And which multiply to give -8
Use -8 and 1, so -7x can be split into -8x + x
x2 - 7x – 8
= x2 + x - 8x – 8
= x(x +1) – 8(x + 1)
= (x - 8)(x +1)
Example
Factorise 6x2 + 23x + 10
Two numbers are needed which add to give 23
And which multiply to give 60
Use 3 and 20, so 23x can be split into 3x + 20x
6x2 + 23x + 10
= 6x2 + 3x + 20x + 10
= 3x(2x + 1)+10(2x + 1)
= (3x + 10)(2x + 1)
!!Warning !!
Sometimes trial and error is involved.