What method most efficient to use to solve x^2+4x+3=0

My recommendation would be to factor ...

x² + 4x + 3 = 0 factor the left side into two binomials

(x + 3) (x + 1) = 0 set each of these binomials equal to zero and solve

x + 3 = 0 Implies x = - 3

x + 1 = 0 implies x = - 1