Need help with factorisation of (x^2+2x)^2-2(x^2+2x)- 3

[Math235]

In the explanation before you start with the factorization sums the book gives 7 examples on how to complete the following sums one of the explanations I don't understand and it is needed for one of the questions which I also do not understand.

Here is the books explanation:

Factorize: (x^2+2x)^2-2(x^2+2x)- 3
=(x^2+2x+1)(x^2+2x-3) I do not know what happens from the question to this step, I do not know if some of the steps are not shown but other than that I couldn't find a method of getting to this step.
=(x+1)^2(x-1)(x+3) This step I understand, the trinomials are turned into binomials as the last step of factorizing it.


If I figure out how to do this I will be able to do the question they ask because it uses the exact same method.
Thanks in advance

 

[Deleted member 4993]

In the explanation before you start with the factorization sums the book gives 7 examples on how to complete the following sums one of the explanations I don't understand and it is needed for one of the questions which I also do not understand.

Here is the books explanation:

Factorize: (x^2+2x)^2-2(x^2+2x)- 3
=(x^2+2x+1)(x^2+2x-3) I do not know what happens from the question to this step, I do not know if some of the steps are not shown but other than that I couldn't find a method of getting to this step.
=(x+1)^2(x-1)(x+3) This step I understand, the trinomials are turned into binomials as the last step of factorizing it.


If I figure out how to do this I will be able to do the question they ask because it uses the exact same method.
Thanks in advance

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

substitute:

u = x^2 + 2x, then,

(x^2+2x)^2-2(x^2+2x)- 3 = u^2 - 2*u + 3

The above is a quadratic equation in 'u'. Can you factorize it?

 

[Math235]

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

substitute:

u = x^2 + 2x, then,

(x^2+2x)^2-2(x^2+2x)- 3 = u^2 - 2*u + 3

The above is a quadratic equation in 'u'. Can you factorize it?

Thank you for the reply, when you substituted:
u = x^2 + 2x, then, (x^2+2x)^2-2(x^2+2x)- 3 = u^2 - 2*u + 3 how did the -3 become a +3.

My main problem is how they went from
(x^2+2x)^2-2(x^2+2x)- 3
to
(x^2+2x+1)(x^2+2x-3)

I don't fully understand your method if you could please explain I understand the substitution part but not from there on.
Thanks for your time.

 

[stapel]

Thank you for the reply, when you substituted:
u = x^2 + 2x, then, (x^2+2x)^2-2(x^2+2x)- 3 = u^2 - 2*u + 3 how did the -3 become a +3.

A typo, I suspect.

I understand the substitution part but not from there on.

So you understand how to get to here:

. . . . .\(\displaystyle u^2\,-\, 2u\, -\, 3\)

...but you don't know how to factor this. To learn how to factor quadratics (which was supposed to have been covered well before assigning the current exercise), please try here. Once you have studied the lesson, think about the factors of -3 that add up to -2....

 

[Math235]

A typo, I suspect.


So you understand how to get to here:

. . . . .\(\displaystyle u^2\,-\, 2u\, -\, 3\)

...but you don't know how to factor this. To learn how to factor quadratics (which was supposed to have been covered well before assigning the current exercise), please try here. Once you have studied the lesson, think about the factors of -3 that add up to -2....

Thanks so much for the help, should the steps look like this?

u^2-2u-3
=(u+1)(u-3)
=(x^2+2x+1)(x^2+2x-3) here we put x^2+2x back in the place of u
=(x+1)^2(x+3)(x-1)

Then I have the answer the explanation showed.

I may have asked my previous question wrong. I do understand factorization I was not getting the +3 -3 thing which you said may have been a typo. I understand Sabhotosh Khan's method now after checking it again I made mistake adding "I don't fully understand your method if you could please explain I understand the substitution part but not from there on".

Thanks @Sabhotosh Khan and @stapel for your help!