The equation provided is:

$x^2 + 8x + 12 = 0$

We are asked to solve this by factoring. Let's follow the steps:

Step 1: Identify the coefficients.

• $a = 1$ (the coefficient of $x^2$)
• $b = 8$ (the coefficient of $x$)
• $c = 12$ (the constant term)

Step 2: Find two numbers that multiply to $a \times c = 1 \times 12 = 12$, and add to $b = 8$.

The numbers that work are $6$ and $2$, because: $6 \times 2 = 12 \quad \text{and} \quad 6 + 2 = 8$

Step 3: Rewrite the middle term using these numbers.

$x^2 + 6x + 2x + 12 = 0$

Step 4: Factor by grouping.

$x(x + 6) + 2(x + 6) = 0$

Step 5: Factor out the common binomial.

$(x + 2)(x + 6) = 0$

Step 6: Solve for $x$.

$x + 2 = 0 \quad \text{or} \quad x + 6 = 0$ $x = -2 \quad \text{or} \quad x = -6$

Thus, the solutions are: $x = -2 \quad \text{and} \quad x = -6$

Would you like a deeper explanation or have any questions?

Related Questions:

1. How do you verify the solutions to a quadratic equation?
2. What is the significance of factoring in solving quadratic equations?
3. Can all quadratic equations be factored?
4. How do you solve quadratic equations using the quadratic formula?
5. What are the real-life applications of solving quadratic equations?

Tip:

Always check your factored terms by expanding them back to the original equation to ensure accuracy.