The quadratic formula was not used when factoring this polynomial. 

It was done by splitting the middle term:

'-3x = -2x - x'

That is why we have

x^2 - 2x - x + 2 = 

then grouping

= (x^2 - 2x) -(x - 2) =

pulling the common factor 'x' in front of the brackets

= x(x - 2) - (x - 2) =

again pulling the common factor 'x-2' in front of the brackets

= (x - 2)(x - 1)

Another was to solve is by finding the roots of the quadratic equation:

x^2 -3x + 2 = 0

The roots are:

x1 = 2

x2 = 1

Click here to see the step by step solving of the quadratic equation

And then use the formula

x^2 + bx + c = (x - x1)(x - x2)

In our case it will be:

x^2 -3x + 2 = (x - 2)(x - 1)