Question 83933
<pre><font size = 4><b>
3(x–2)<sup>2</sup> – 3(x–2) – 6

To see this better, I will color the parentheses red:

3<font color = "red">(x-2)</font><sup>2</sup> – 3<font color = "red">(x-2)</font> – 6

Look carefully at each of these and you will notice
just how similar they are.  Especially observe the
red parts:

3<font color = "red">(x-2)</font><sup>2</sup> – 3<font color = "red">(x-2)</font> – 6       and      3<font color = "red">y</font><sup>2</sup> - 3<font color = "red">y</font> - 6          

Do you see that the only difference between them is
their <font color = "red">red</font> parts?  The expression on the right is
an ordinary factoring job, and we do it this way:

                                         3y<sup>2</sup> - 3<font color = "red">y</font> - 6
                                        3(y<sup>2</sup> - <font color = "red">y</font> - 2)  
                                       3(<font color = "red">y</font> - 2)(<font color = "red">y</font> + 1)

So we do the exact same thing with your problem:

 3<font color = "red">(x-2)</font><sup>2</sup> – 3<font color = "red">(x-2)</font> – 6        --->        3<font color = "red">y</font><sup>2</sup> - 3<font color = "red">y</font> - 6
3[<font color = "red">(x-2)</font><sup>2</sup> - <font color = "red">(x-2)</font> - 2]        --->       3(<font color = "red">y</font><sup>2</sup> - <font color = "red">y</font> - 2)
3[<font color = "red">(x-2)</font> - 2][<font color = "red">(x-2)</font> + 1]      --->      3(<font color = "red">y</font> - 2)(<font color = "red">y</font> + 1)

Do you see that what's done on the left is just like
what's done on the right except that on the right we
are doing it with just a single letter <font color = "red">y</font>, whereas on
the left we are doing those same steps only with 
something a bit more complicated-looking, namely <font color = "red">(x-2)</font>. 
We're also using brackets [ ] on your problem because
the red expression already contains parentheses <font color = "red">( )</font>.

This is a skill of algebra that you must develop --

Viewing a big chunk of algebra like <font color = "red">(x-2)</font> just as you
view a small chunk of algebra, such as <font color = "red">y</font>.

There is only a couple things more that we need to do with
your problem than we had to do with the one on the right. 
They are to remove the parentheses inside the brackets,
collect terms, and change the brackets to parentheses.

3[(x-2) - 2][(x-2) + 1]
 3[ x-2 - 2][x-2 + 1]
     3[x-4][x-1]
     3(x-4)(x-1) 

Edwin</pre>