6x²-7x-3

Multiply the green 6 times the red -3.
That gives -18.
Think of two integers which have
   1. Product = -18
   2. Sum = the blue -7
Such a pair of integers are -9 and +2
because 
   1. Product = (-9)x(+2) = -18 
   2. Sum =     (-9)+(+2) = -7

Replace the middle term -7x by -9x + 2x, using 
the -9 and the +2 as coefficients of x.

6x² - 9x + 2x - 3

Factor the first two terms by taking out GCF 3x

  6x² - 9x + 2x - 3
3x(2x - 3) + 2x - 3

Factor the last two terms by taking out GCF = +1
(When there is nothing bigger that can be factored out,
then factor out +1 if the next to last term is 
positive, or -1 if it is negative.)

3x(2x - 3) + 2x - 3
3x(2x - 3) + 1(2x - 3)

Now notice that the red expression (2x - 3) appears
in both terms as shown by the colors below:

3x(2x - 3) + 1(2x - 3)

So factor out the red expression (2x - 3) and
place the black parts inside parentheses:

3x(2x - 3) + 1(2x - 3)

(2x - 3)(3x + 1)

That's it, and you can just make it all black:

(2x - 3)(3x + 1)

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That's the answer. You can check it by FOIL:

          6x² + 2x - 9x - 3

and combining the middle two terms:

          6x² - 7x - 3

which is what we started with, so 

(2x - 3)(3x + 1) is the correct factorization.

Edwin