To simplify the expression $\frac{x^2 + 4x - 12}{2x + 12}$, follow these steps:

1. Factor the numerator and denominator:

   • Numerator: $x^2 + 4x - 12$
     Factor $x^2 + 4x - 12$. Find two numbers that multiply to $-12$ and add to $4$. These numbers are $6$ and $-2$, so: $x^2 + 4x - 12 = (x + 6)(x - 2)$

   • Denominator: $2x + 12$
     Factor out the greatest common factor (GCF), which is $2$: $2x + 12 = 2(x + 6)$

2. Simplify the expression: $\frac{(x + 6)(x - 2)}{2(x + 6)}$ Since $x + 6$ is common in both the numerator and the denominator, cancel it out: $\frac{x - 2}{2}$

So, the simplified form of the expression is:

$\frac{x - 2}{2}$

Would you like further clarification or more details on any step?

Here are some related questions to expand on this concept:

1. How do you factor quadratic expressions like $x^2 + 4x - 12$?
2. What is the greatest common factor (GCF) and how is it used in factoring?
3. How do you cancel terms when simplifying fractions?
4. What are the conditions for a rational expression to be undefined?
5. Can we simplify a fraction when the numerator and denominator contain terms that cannot be factored?

Tip: Always check for common factors before simplifying fractions.