Question 6213
Since you are given a choice of methods, you can see (or should be able to) that "factorisation" won't work here so you would use "completing the square".

x^2 + 2x + 3 = 0  Subtract 3 from both sides.
x^2 + 2x = -3 Complete the square on the left by adding the square of half the coefficient of the x-term to both sides. (Half of 2 is 1.  1 squared is 1)

x^2 + 2x + 1 = -3 + 1
x^2 + 2x + 1 = -2 Factor the left side.
(x + 1)(x + 1) = -2
(x + 1)^2 = -2 Take the square root of both sides.
x+1= +/- sqrt(-2)
x = -1 +/- sqrt(-2)
x = -1 + sqrt(2)i or x = -1 - sqrt(2)i  Where i = sqrt(-1)