To solve the quadratic equation \(x^2 + 8x + 12 = 0\), we can use the quadratic formula:

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

Here, the coefficients are:
- \(a = 1\) (coefficient of \(x^2\))
- \(b = 8\) (coefficient of \(x\))
- \(c = 12\) (constant term)

Substituting these values into the quadratic formula:

\[
x = \frac{-8 \pm \sqrt{8^2 - 4(1)(12)}}{2(1)}
\]

\[
x = \frac{-8 \pm \sqrt{64 - 48}}{2}
\]

\[
x = \frac{-8 \pm \sqrt{16}}{2}
\]

\[
x = \frac{-8 \pm 4}{2}
\]

This gives us two possible solutions:

1. \(x = \frac{-8 + 4}{2} = \frac{-4}{2} = -2\)
2. \(x = \frac{-8 - 4}{2} = \frac{-12}{2} = -6\)

So, the solutions are \(x = -2\) and \(x = -6\).

Would you like more details or have any questions about this solution?

Here are five related questions to expand on this topic:
1. How can you factor the quadratic equation \(x^2 + 8x + 12 = 0\) directly?
2. What is the discriminant, and how does it help determine the nature of the roots of a quadratic equation?
3. How can completing the square be used to solve the quadratic equation \(x^2 + 8x + 12\)?
4. What are the graphical interpretations of the solutions to \(x^2 + 8x + 12 = 0\)?
5. How does the quadratic equation change if we vary the coefficient \(b\) while keeping \(a = 1\) and \(c = 12\)?

**Tip:** Always check your solutions by substituting them back into the original equation to ensure they satisfy the equation.

Learn how to solve the quadratic equation x^2 + 8x + 12 = 0 with a detailed step-by-step solution. This problem covers key algebraic concepts such as the quadratic formula and the discriminant. Suitable for high school students in Grades 10-12.

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