4n^2 + 12 n + 5 = 0

it is always better to make coefficient of square term as 1. So divide the equation by 4, we get

n^2 + 3n + 5/4 = 0

n^2 + 3n = - 5/4

Add square of half the coefficient of n ( in this case ( 3/2)^2= 9/4 ) on both sides

n^2 + 3n + ( 3/2 )^2= 9/4 - 5/4

[n + 3/2]^2 = 1 that gives n + 3/2 = 1 or -1 that gives n = -1/2 OR -5/2