Solve this equation and select the choice that contains all of the solutions. \[ (x+7)^{2}=-16 \]

First, we notice that the equation \((x+7)^{2}=-16\) presents a challenge because the square of any real number is non-negative, and thus it cannot equal a negative number like \(-16\). This means we will be working in the realm of complex numbers. To solve for \(x\), we start by taking the square root of both sides: \[ x + 7 = \pm \sqrt{-16}. \] The square root of \(-16\) can be simplified as: \[ \sqrt{-16} = 4i, \] where \(i\) is the imaginary unit. Thus, we have: \[ x + 7 = \pm 4i. \] To isolate \(x\), we can subtract 7 from both sides: \[ x = -7 \pm 4i. \] This gives us two solutions: \[ x = -7 + 4i \quad \text{and} \quad x = -7 - 4i. \] So the complete set of solutions is: \[ \{-7 + 4i, -7 - 4i\}. \]