We want to factorise 2x^2-21x-11
We look for two numbers which add to give the coefficient of x; so we seek two numbers which add to give -21.
However, as the coefficient of x^2 is not 1 then instead of looking for two numbers which multiply to give −11 we must look for two numbers which multiply to give −22, (that is, the coefficient of x^2 multiplied by the constant term, 2 × −11.
a × b = -22
a + b = -21
By inspection (or trial ad error) we can find two numbers a=-22 and b=1
So we have,
2x^2-21x-11 = 2x^2-22x + x-11
:. 2x^2-21x-11 = 2x(x-11) + x-11 (by factorising the first two terms)
:. 2x^2-21x-11 = 2x(x-11) + (x-11) (collecting common terms)
:. 2x^2-21x-11 = (x-11)(2x+1) (by factorising the last two terms)
We can check this by multiplying out:
(x-11)(2x+1) = 2x^2 + x -22 x -11 = 2x^2-21x-11 QED
