firstly, you'll want to get rid of the 4

\(\displaystyle 4x^2+16x -12 = 0\)

\(\displaystyle x^2+4x-3 = 0\)

Then, ask yourself, if you work out the brackets of \(\displaystyle (x+c)^2\), what makes sure you get +4x ? In this case that is \(\displaystyle c=2 \frac{4}{2}\), because \(\displaystyle (x+2)^2 = x^2 + 4x + 4\)

So, what you have now is:

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

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

\(\displaystyle (x+2)^2 -7 = x^2 + 4x - 3\)

Therefore, \(\displaystyle (x+2)^2 = 7\) is equivalent to the first equation.

Does that make it clear?