Question

How do you factor `3x^2( 4x-12)^2 + x^3(2)(4x-12)(4)`?

Answer 1

`8x^2(x-3)(5x-7)`
This is more of a guide to approach than deriving the correct answer. You can check that!!!!!

Explanation:

The trick is to look for common elements that can be factored out and then playing around until you find the solution.
Looking at the question you will observe that `x^2(4x-12)` is a common factor. So let us 'play' with that and see what we get.
I am taking it one step at a time so that you can see the process and develop your own faster approaches.

Consider `3x^2(4x-12)^2`
this can be factored so that you have:

`x^2(4x-12) times 3(4x-12)` ..............................( 1 )

Consider `x^3(2)(4x-12)(4)`
This can be factored so that you have:

`x^2(4x-12) times 8x`.............................................( 2 )

Putting ( 1 ) and ( 2 ) together gives:

`x^2(4x-12) times 3(4x-12) + x^2(4x-12) times 8x`

Factoring again

`x^2(4x-12)(3[4x-12] + 8x)`

`x^2(4x-12)(12x-36+8x)`

`x^2(4x-12)(20x-36)`

Notice that all the numbers in the bracket are even. This means that we can take it 'down' again by factor of 2

`2x^2(2x-6) times 2(10x - 18)`

Again they are all even within the brackets so let us repeat the process:

`4x^2(x-3) times 4(5x - 9)`

But `4x^2` from `4x^2(x-3)` multiplied by 4 from `4(5x-9)` = `8x^2`

Substituting this in gives:

`8x^2(x-3)(5x-9)`

I have not checked that the final answer is correct. I will leave that to you to do!!! However you can see the process types available to you!!!