To solve the equation:

$x^2 + 7x + 12 = 0$

we can use factoring.

Step-by-Step Solution:

1. Identify the equation: We have a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 1$, $b = 7$, and $c = 12$.

2. Look for factors of 12 that add up to 7:

   • The factors of 12 are $(3, 4)$, since $3 + 4 = 7$.

3. Rewrite the equation using these factors: $x^2 + 7x + 12 = (x + 3)(x + 4) = 0$

4. Set each factor equal to zero:

   • $x + 3 = 0$ gives $x = -3$.
   • $x + 4 = 0$ gives $x = -4$.

Solution:

The solutions to the equation are: $x = -3 \quad \text{and} \quad x = -4$

Would you like further details or have any questions?

Here are some related questions for further exploration:

1. How do we solve quadratic equations when factoring is not possible?
2. What is the quadratic formula, and how does it work?
3. How can we verify the solutions of a quadratic equation?
4. What are real-world applications of quadratic equations?
5. How does the discriminant in the quadratic formula help in determining the nature of the roots?

Tip: Always double-check that the factors multiply back to the original equation to confirm accuracy.