The question requires solving two parts:

(a) Fully factorize $x^2 + 7x + 12$:

We find two numbers that multiply to $12$ (the constant term) and add up to $7$ (the coefficient of $x$):

• The numbers are $3$ and $4$ because $3 \cdot 4 = 12$ and $3 + 4 = 7$.

Thus, the factorization is: $x^2 + 7x + 12 = (x + 3)(x + 4)$

(b) Use your answer to part (a) to solve $x^2 + 7x + 12 = 0$:

We set the factorized form equal to zero: $(x + 3)(x + 4) = 0$

Solve each factor for $x$:

1. $x + 3 = 0 \implies x = -3$
2. $x + 4 = 0 \implies x = -4$

Thus, the solutions are: $x = -3 \quad \text{and} \quad x = -4$

Let me know if you'd like a more detailed breakdown or have additional questions!

Here are 5 related questions to explore further:

1. How do you identify numbers to factorize a quadratic equation quickly?
2. What is the difference between factorization and solving a quadratic equation?
3. Can you solve $x^2 - 5x + 6 = 0$ using the same method?
4. How can you verify that your factorization is correct?
5. What if a quadratic equation cannot be factorized neatly?

Tip: Always check your factors by expanding them back to ensure accuracy!