I am working on factoring polynomials and have an example: I understand the basi

Question

I am working on factoring polynomials and have an example:   I understand the basics of factoring but there is one issue I am having difficulty with.   The problem is as follows:

Factor as completely as possible:

3X3 - 3X2 - 18

The solution that the text gives me is as follows:

3X(X2-X-6)

3X [X2 -X+(1/2)2 -6-(1/2)2]

3X[(x-1/2)2 - (5/2)2]

3x(x-1/2+5/2)(x-1/2-5/2)

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

The issue that I am having is how do they obtain the exponential factors, and what causes them to be placed in the positions that they are in . i.e. (1)2, (5/2)2.    I can see that the answer proves when applying FOIL.   I am not confident on how the answer is derived through this method. Can you clarify this method for me.  

Explanation / Answer

Yes, the following of the text is correct, or you can factor 3 first, then do a factoring Anyway, gets same answer

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