Final answer:
After taking the square root of both sides of the equation, we find that x equals -2 + 4i or -2 - 4i, which are complex numbers.
Step-by-step explanation:
In solving the equation \((x+2)^2 = -16\), we start by taking the square root of both sides to isolate \(x\). Upon taking the square root of a negative number, we end up with an imaginary number. Hence, x + 2 = \pm 4i, which results in two possible solutions for \(x\) after subtracting 2 from both sides: x = -2 + 4i or x = -2 - 4i.
These complex solutions emerge because the square of a real number is never negative, and \(i\) represents the square root of -1.