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The equation provided is:

\[
n^2 + 7n + 12 = 0
\]

To solve this quadratic equation, we will try factoring it. We are looking for two numbers that multiply to 12 (the constant term) and add up to 7 (the coefficient of the middle term).

The numbers 3 and 4 satisfy this condition since:

\[
3 \times 4 = 12 \quad \text{and} \quad 3 + 4 = 7
\]

Thus, we can factor the quadratic as:

\[
(n + 3)(n + 4) = 0
\]

Now, we set each factor equal to zero:

\[
n + 3 = 0 \quad \text{or} \quad n + 4 = 0
\]

Solving for \( n \), we get:

\[
n = -3 \quad \text{or} \quad n = -4
\]

Therefore, the solutions are:

\[
n = -3 \quad \text{and} \quad n = -4
\]

Would you like more details on the factoring method, or do you have any questions? Here are some related questions:

1. What is the quadratic formula, and how can it be applied to this equation?
2. How can we confirm the correctness of the solutions?
3. What are the conditions for factoring quadratic equations?
4. How does the discriminant help in determining the nature of the roots?
5. Can all quadratic equations be factored easily?

**Tip:** Factoring is one of the quickest methods for solving quadratic equations when the factors are integers.

Learn how to solve the quadratic equation n^2 + 7n + 12 = 0 by factoring. This algebra problem demonstrates factoring techniques and solving for n. Suitable for students in Grades 8-10.

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