What we do is take half of the x coefficient, which in this case is 4. So take half of 4 to get 2. Then square 2 to get 2^2 = 4.
We will add and subtract 4 to the expression so that things don't change. It's effectively the same as adding zero.
Once this happens, you rearrange terms and factor.
x^2 + 4x + 3
x^2 + 4x + 3 + 0
x^2 + 4x + 3 + (4-4)
(x^2 + 4x + 4) + 3 - 4
(x^2 + 4x + 4) - 1
(x+2)^2 - 1
Answer: (x+2)^2 - 1
