To solve the quadratic equation \(x^2 + 4x + 16 = 0\), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = 4\), and \(c = 16\). Let's calculate the discriminant first:

\[
b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 16 = 16 - 64 = -48
\]

Since the discriminant is negative, there are no real solutions; instead, there are two complex solutions.

Now, we can find the solutions:

\[
x = \frac{-4 \pm \sqrt{-48}}{2 \cdot 1}
\]
\[
x = \frac{-4 \pm \sqrt{48}i}{2}
\]
\[
x = \frac{-4 \pm 4\sqrt{3}i}{2}
\]
\[
x = -2 \pm 2\sqrt{3}i
\]

Thus, the solutions are:

\[
x = -2 + 2\sqrt{3}i \quad \text{and} \quad x = -2 - 2\sqrt{3}i
\]

If you have any questions or need further clarification, feel free to ask! Here are some related questions you might consider:

1. What is the significance of the discriminant in quadratic equations?
2. How would you graph the quadratic function \(y = x^2 + 4x + 16\)?
3. Can you explain the process of completing the square for this equation?
4. What are the properties of complex numbers involved in the solutions?
5. How would the solutions change if the constant term were different?

**Tip:** Always check the discriminant to determine the nature of the roots in a quadratic equation.

Explore the step-by-step solution to the quadratic equation x^2 + 4x + 16 = 0. Understand the use of the quadratic formula and complex numbers in finding solutions when the discriminant is negative. This problem is ideal for students in Grades 9-11.

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