The equation given is:

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

We need to solve this by factoring. The goal is to factor the quadratic expression into two binomials. We are looking for two numbers that multiply to give the constant term $12$ and add to give the coefficient of the middle term $-7$.

The pair of numbers that satisfy this condition are $-3$ and $-4$, since:

$(-3) \times (-4) = 12 \quad \text{and} \quad (-3) + (-4) = -7$

Thus, the factored form of the equation is:

$(x - 3)(x - 4) = 0$

Setting each factor equal to zero gives:

$x - 3 = 0 \quad \text{or} \quad x - 4 = 0$

So, the solutions are:

$x = 3 \quad \text{and} \quad x = 4$

Since the problem asks for the smallest solution first, the answer is:

$x = 3, 4$

Would you like more details, or have any further questions?

Here are 5 related questions you could explore:

1. How would the solution change if the equation was $x^2 + 7x + 12 = 0$?
2. Can you solve a quadratic equation by completing the square instead of factoring?
3. What are the conditions for a quadratic equation to have no real solutions?
4. How can you check if the factoring was done correctly?
5. How do you apply the quadratic formula to solve this equation?

Tip: Always check your factored form by multiplying it back out to verify it equals the original quadratic equation.