3x^2(4x-12)^2 + x^3(2)(4x-12)(4) HALP

[Hierophant]

3x^2(4x-12)^2 + x^3(2)(4x-12)(4)
Factor this expression completely. This type of question occurs in calculus in using the "product rule".

Attempt:
I first factor out a x^2 (4x-12) factor out the 4 from this, then have 4x^2 (X-3)
The left overs are: 3(4x-12) + x(2)(4)
I factor out a 4 from the left overs, I now have. 16x^2(x-3) 3(x-3) + x(2)
Then I simplify: 16x^2 (x-3) 3(x-3) +2x = 16x^2 (x-3) (5x-9)


Which is the right answer, but I may have been biased into finding this as I did know the answer before hand. So is my reasoning sound? Maybe you could go through this yourself and show me your process, practice does not hurt :)

 

[UltrafastPED]

When you factor out x^2 (4x-12) you have:
[x^2 (4x-12)]*[3(4x-12) + 8x] ... which is what you got. Then you factor out the 4:
[16 x^2 (x-13)]*[3(x-3)+2x] => [16 x^2 (x-3)]*[3x - 9 +2x] => [16 x^2 (x-3)]*[5x-9]
And then discard the extra brackets.

Like you, I did not notice the extra factor of 4 in (4x-12) on the first pass.

 

[Hierophant]

Thanks for the reply. I have another question...

Factor this completely.

4a^2c^2 - (a^2 - b^2 + c^2)^2

I am so stumped on this one.

Thanks again for the previous answer.

 

[UltrafastPED]

Square the second term ... you should get six terms. One of these six can be combined with the first term of the original expression.

Then look for simplifications ...

 

[epenguin]

Thanks for the reply. I have another question...

Factor this completely.

4a^2c^2 - (a^2 - b^2 + c^2)^2

I am so stumped on this one.

Thanks again for the previous answer.

You often get things like this in exercises and even sometimes in real science problems. This is a "difference of two squares". You've probably seen the factorising of that. After using that there might be some further simplification possible.