SOLUTION: Factor the expression completely: x^3(x-3)^2 (x-1) - 2x^2(x-3)^2

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Question 409454: Factor the expression completely:
x^3(x-3)^2 (x-1) - 2x^2(x-3)^2
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E3%28x-3%29%5E2++%28x-1%29++-+2x%5E2%28x-3%29%5E2
The most straightforward (but extremely slow) way to factor this would be to multiply this out first, combine the common terms, if any, and then try to factor the simplified expression.

The faster way is based on seeing that there is a common factor to the expression you were given. I have highlighted it in red below:
x%5E3red%28%28x-3%29%5E2%29++%28x-1%29++-+2x%5E2red%28%28x-3%29%5E2%29
This common factor can be factored out:
red%28%28x-3%29%5E2%29%28x%5E3%28x-1%29++-+2x%5E2%29
In the second factor there is a common factor of x%5E2 which we can factor out:
%28x-3%29%5E2%2Ax%5E2%28x%28x-1%29++-+2%29
The third factor can now be simplified:
%28x-3%29%5E2%2Ax%5E2%28x%5E2-x+-+2%29
The third factor can be factored:
%28x-3%29%5E2%2Ax%5E2%28x%2B1%29%28x+-+2%29
And we're done.