The equation given is:

$x^{2} + 16 = 0$

To solve this, first isolate $x^{2}$ by subtracting 16 from both sides:

$x^{2} = -16$

Next, take the square root of both sides. Keep in mind that the square root of a negative number involves imaginary numbers. Therefore:

$x = \pm \sqrt{-16}$

We can express $\sqrt{-16}$ as $\sqrt{-1} \times \sqrt{16}$, and $\sqrt{-1}$ is the imaginary unit $i$. Thus:

$x = \pm 4i$

So, the solutions to the equation are:

$x = 4i \quad \text{or} \quad x = -4i$

These are the two complex solutions.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you solve quadratic equations with real solutions?
2. What are imaginary numbers, and how are they used in mathematics?
3. Can you solve an equation with higher powers involving imaginary numbers?
4. What are complex numbers, and how do you add or multiply them?
5. How would this solution change if we had $x^2 - 16 = 0$?

Tip: When solving equations with negative square roots, always check if imaginary numbers are involved (use $i = \sqrt{-1}$).