well you are almost to the solution:

convert the base to 10

log10 (x^2 + 4x + 3) / log10 3 = 1

log10 (x^2 + 4x + 3) = log10 3

comparing left and right hand side:

x^2 + 4x + 3 = 3

solving for x you will get x = 0, x = -4

then you substitute for each value of x into the main equation and carry out your reasoning why which value of x is acceptable and which is not acceptable