[Mike Betts]

OK, so now I'm feeling old... We used to call this "completing the square".

Your first question: Multiply it out...

y=2(x+1)^2-3
y=2[(x+1)(x+1)]-3
y=2[x^2+2x+1]-3
y=2x^2+4x+2-3
y=2x^2+4x-1

Now for the reverse (completing the square). First, make a=1 and (temporarily) get rid of c.

y=2x^2+4x-1
y=2[x^2+2x]-1 ***

Now, we concentrate on the equation in brackets. What would be have to add/subtract to factor this?

[x^2+2x] ==> [x^2+2x+1] = [(x+1)(x+1)]

Note that we added 1 to complete the square. Now:

[x^2+2x] = [(x^2+2x+1)-1] = [(x+1)(x+1)-1]

And we substitute the bracketed equation back into *** as follows:

y=2[x^2+2x]-1 (I repeated *** for clarity)
y=2[(x+1)(x+1)-1]-1
y=[2(x+1)^2-2]-1
y=2(x+1)^2-3

Does this make more sense?