Multiple Choice 1 point Solve \( (x+2)^{2}=-16 \) \( 2,-6 \) \( \pm 3 i \sqrt{2} \) \( 2 \pm 4 i \) \( -2 \pm 4 i \)

To solve the equation \( (x+2)^{2} = -16 \), we begin by taking the square root of both sides, leading us to: \[ x + 2 = \pm 4 i \] Next, we isolate \( x \) by subtracting 2 from both sides, giving us: \[ x = -2 \pm 4 i \] So the correct answer is \( -2 \pm 4 i \). In the world of complex numbers, the term \( i \) represents the imaginary unit, defined as the square root of -1. This concept has allowed mathematicians to solve equations that have no real solutions, expanding our understanding of mathematics beyond the realm of real numbers. Complex numbers are hugely beneficial in fields like engineering and physics, where wave functions and oscillations often involve imaginary and real parts. In many algebraic problems, students often fall into the trap of trying to solve equations like \( (x+2)^2 = -16 \) by taking square roots and ignoring the implications of negative results under the square root. It's important to remember that the presence of a negative number will lead to complex solutions, which can be a bit mind-bending at first! Staying mindful of these details helps avoid common mistakes and leads to a clearer understanding of the subject.