$\displaystyle f(x) = 2x^2+8x+7$

$\displaystyle = 2(x^2+4x)+7$

$\displaystyle =2(x^2+4x+4-4)+7$

Why are we subtracting here?

Continuing...it says though before what's below:

After adding and subtracting 4 within the parentheses, you must now regroup the terms to form a perfect square trinomial. The -4 can be removed from inside the parentheses (

4-4 is zero...why is the -4 there, what is the point if there's still going to be a 4 there somehow untouched); however, because of the 2 outside of the parantheses (

where did that come from?), you must multiply by -4 by 2, as shown below.

$\displaystyle f(x) = 2(x^2+4x+4) -2(4)+7$

$\displaystyle = 2(x^2+4x+4)-8+7$

$\displaystyle = 2(x+2)^2-1$