Question 957249
<pre>
She's right BUT!!! -- how in the lulu was she 
able to go from

this step:

{{{c^2=4x^4+8x^3+8x^2+4x+1}}}

to this step????:

{{{c^2=(2x^2+2x+1)^2}}}

It is correct but she doesn't explain how she did that.
I have no idea how she did that myself!  A magic wand!  :)

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I would not have multiplied everything out as she did.
Here's what I would have done:

{{{c^2=(2x(x+1)^"")^2+(2x+1)^2}}}

I'd multiply {{{(2x+1)^2}}} out as {{{4x^2+4x+1}}}

{{{c^2=(2x(x+1)^"")^2+4x^2+4x+1}}}

Then I'd realize that if we factor {{{4x}}} out of {{{4x^2+4x}}}
we get {{{4x(x+1)}}} which is 2 times the 2x(x+1) in the 
parentheses in the first term on the right:

{{{c^2=(2x(x+1)^"")^2+4x(x+1)+1}}}

Now to make things simpler, I will let u = 2x(x+1)
then 4x(x+1) will equal to 2u, and we have:

{{{c^2=u^2+2u+1}}

which factors as 

{{{c^2=(u+1)^2}}}

Taking positive square roots

{{{c=u+1}}}

And since {{{u = 2x(x+1)}}}, we have:

{{{c = 2x(x+1)+1}}}

{{{c = 2x^2+2x+1}}}

Edwin</pre>