We want to factorise 4x^2-8x-12+6x = 4x^2-2x-12
We look for two numbers which add to give the coefficient of x; so we seek two numbers which add to give -2.
However, as the coefficient of x^2 is not 1 then instead of looking for two numbers which multiply to give the constant term -12 we must look for two numbers which multiply to give -48, (that is, the coefficient of x^2 multiplied by the constant term, ie 4 × −12.
a × b = -48
a + b = -2
By inspection (or trial and error) we can find two numbers a=-8 and b=6
So we have,
4x^2-2x-12 = 4x^2-8x + 6x-12
:. 4x^2-2x-12 = 4x(x-2) + 6x-12 (by factorising the first two terms)
:. 4x^2-2x-12 = 4x(x-2) + 6(x-2) (collecting common terms)
:. 4x^2-2x-12 = (4x+6)(x-2) (by factorising the last two terms)
