The problem asks to factor the quadratic expression $7x^2 + 12x - 4$ by grouping, also called the AC-method.

Step-by-Step Solution:

1. Identify the coefficients:

   • $a = 7$ (coefficient of $x^2$)
   • $b = 12$ (coefficient of $x$)
   • $c = -4$ (constant)

2. Multiply $a$ and $c$: Multiply $7 \times -4 = -28$.

3. Find two numbers that multiply to $-28$ and add to $12$: The two numbers are $14$ and $-2$ since:

   • $14 \times (-2) = -28$
   • $14 + (-2) = 12$

4. Rewrite the middle term $12x$ using these two numbers: Rewrite $7x^2 + 12x - 4$ as: $7x^2 + 14x - 2x - 4$

5. Factor by grouping: Group the terms in pairs: $(7x^2 + 14x) - (2x + 4)$ Factor out the common terms: $7x(x + 2) - 2(x + 2)$

6. Factor the common binomial $x + 2$: $(7x - 2)(x + 2)$

Final Factorization:

$7x^2 + 12x - 4 = (7x - 2)(x + 2)$

This would be the correct factorization to input.

Let me know if you need further clarification or additional details.

Here are five related questions to expand your understanding:

1. What is the AC-method, and why is it useful for factoring quadratic expressions?
2. How do you factor a quadratic expression when the leading coefficient is not 1?
3. Can every quadratic expression be factored by grouping? Why or why not?
4. How would you handle a situation where the quadratic cannot be factored by integers?
5. How does factoring by grouping differ from using the quadratic formula?

Tip: Always check if you can factor a common term from all parts of the quadratic before applying the AC-method. This simplifies the process.