Middle term problem in a trinomial

Question

In a trinomial when finding the

(two numbers) of the product of the leading coefficient and constant, a×c , and that the sum is the middle term, why is it that the (two numbers) replace the middle term?

For ex: 6x^2 + 4x -2

The (two numbers) are 6 and -2

Question: Why is it that these two numbers are aligned with the variable X and replace the middle term?

I know that they create a four term polynomial that can factored by grouping but there has to be more meaning.

Answer 1

6x^2 + 4x - 2

= 2(3x^2 +2x -1)

=2(3x-1)(x+1)

= either (6x-2)(x+1) or (3x-1)(2x+2)

using FOIL (First terms multiplied + Outside terms multiplied + Inside terms multiplied + Last terms multiplied) = 6x^2 + 6x -2x -2

= 6x^2 +4x -2

the coefficient of the x^2 term added to the constant term = 6-2 =4 which is the coefficient of the middle term

but that's a coincidence, and not generally true for other trinomials.