Question

O.K. my question is as follows ; how do you go from x ^ 2+ 4 to the answer -2i?

Answer 1

You mean from x^2 + 4 = 0 to x = +- 2i?

Answer 2

If you have an equation of the form x^2 = b, the solution is x = +-sqrt(b)
For example, x^2 = 9, then x = 3 or x = -3
x^2 = 17, then x = sqrt(17) or x = -sqrt(17)
In your case, you have x^2 + 4 = 0
Subtract 4 from both sides:
x^2 = -4
This is an eq of the form x^2 = b, with b = -4, so
x = sqrt(-4) or x = -sqrt(-4)
x = i*sqrt(4) or x = -i*sqrt(4)
x = 2i or x = -2i