How to simplify this expression?

[galabingo]

Hi,
I have a problem with simplifying 4(2x+1)(3x-1)^4 + 12(3x-1)^3(2x+1)^2
I know that the answer is 4(9x+2)(2x+1)(3x-1)^3
I will be very thankful for an explanation

 

[Dr.Peterson]

I'd start by recognizing that both terms contain a factor of 4, and at least one factor of (2x+1) and at least three factors of (3x-1), so I'd factor that GCF out:

4(2x+1)(3x-1)^3[...].​

Can you finish that step, so we can see how well you do this far?

Ideally, we want to see all the work you did on your own, to see whether you went in the wrong direction, or just made little mistakes along the way.

 

[galabingo]

So it should be
4(2x+1)(3x-1)^3 x (3x-1)
And
4(2x+1)(3x-1) x 3(2x+1)(3x-1)^2
4(2x+1)(3x-1) x ((3x-1)+3(2x+1)(3x-1)^2)
Therefore
((3x-1)+3(2x+1)(3x-1)^2)= (3x-1) +(6x+3)(3x-1)^2)
=(3x-1) +(6x+3)(9x^2-6x+1)
=(3x-1) + (27x^3+36x^2+6x +27x^2-18x+3)
=3x-1+27x^3+36x^2+6x +27x^2-18x+3
=-9x +27x^3+63x^2-12x+3
=-21x+27x^3+63x^2+3
=-7x+9x^3+21x^2+1
Should I then use long division?
Not sure what to do, I guess it is easy as simplifying was not the main point of the exercise I’ve been doing but I’m still confused

 

[Dr.Peterson]

PLEASE don't use "x" to mean both a variable and multiplication! I can't read those first few lines. I don't think you factored the same thing out of both terms; the GCF includes an exponent of 3, which you lost.

But the important thing is that you must not distribute (expand) everything. There was a reason we factored out the GCF at the start! (Did you not compare what I said to do with the form given for the answer? Factors are your friends, here, so you don't want to send them away.) So everything after "therefore" is going in the wrong direction; you just need to correct the part above that, and then expand only what's inside my [ ... ].

 

[Narwalalpaca12$]

PLEASE don't use "x" to mean both a variable and multiplication! I can't read those first few lines. I don't think you factored the same thing out of both terms; the GCF includes an exponent of 3, which you lost.

But the important thing is that you must not distribute (expand) everything. There was a reason we factored out the GCF at the start! (Did you not compare what I said to do with the form given for the answer? Factors are your friends, here, so you don't want to send them away.) So everything after "therefore" is going in the wrong direction; you just need to correct the part above that, and then expand only what's inside my [ ... ].

You could use either a different variable but in general I would use a * or a dot. Even if x isn’t a variable, people might mistaken it to be and x