The given equation is:

\[
x^2 + 8 + 12 = 0
\]

First, simplify the constant terms on the left-hand side:

\[
x^2 + 20 = 0
\]

Now, subtract 20 from both sides:

\[
x^2 = -20
\]

Next, take the square root of both sides. Since the square root of a negative number introduces an imaginary component, we have:

\[
x = \pm \sqrt{-20} = \pm \sqrt{20}i = \pm 2\sqrt{5}i
\]

So the solution to the equation is:

\[
x = \pm 2\sqrt{5}i
\]

This means there are two complex solutions:

\[
x = 2\sqrt{5}i \quad \text{or} \quad x = -2\sqrt{5}i
\]

Would you like further clarification or details?

Here are 5 related questions you might want to explore:
1. How do you solve quadratic equations with real roots?
2. What are complex numbers and how do they work?
3. Can all quadratic equations have imaginary solutions?
4. What is the significance of the imaginary unit \(i\)?
5. How do you simplify square roots of negative numbers?

**Tip**: When solving quadratic equations, always check for negative roots under the square root, as they will result in imaginary numbers.

Learn how to solve the quadratic equation x² + 8 + 12 = 0, leading to complex solutions. This step-by-step guide covers key algebraic concepts, including handling imaginary numbers and square roots of negative values. Perfect for students in Grades 9-10.

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