Question 104536
Please help me factor the following: 
6x²-7x-3
<pre><font size = 5><b>
<font color = "darkgreen">6</font>x²<font color = "darkblue">-7</font>x<font color = "red">-3</font>

Multiply the <font color = "darkgreen">green 6</font> times the <font color = "red">red -3</font>.
That gives -18.
Think of two integers which have
   1. Product = -18
   2. Sum = the <font color = "darkblue">blue -7</font>
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 <font color = "darkblue">-7x</font> by <font color = "darkblue">-9x + 2x</font>, using 
the -9 and the +2 as coefficients of x.

6x² <font color = "darkblue">- 9x + 2x</font> - 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 <font color = "red">red expression (2x - 3)</font> appears
in both terms as shown by the colors below:

3x<font color = "red">(2x - 3)</font> + 1<font color = "red">(2x - 3)</font>

So factor out the <font color = "red">red expression (2x - 3)</font> and
place the black parts inside parentheses:

3x<font color = "red">(2x - 3)</font> + 1<font color = "red">(2x - 3)</font>

<font color = "red">(2x - 3)</font>(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</pre>