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.