How does the 'splitting the middle term' technique work?

From my other site
http://www.quora.com/Algebra/How-does-the-splitting-the-middle-term-technique-work

Algebra students are taught to factor quadratics using a technique called "splitting the middle term." It is a wonderful technique designed to work for all those problems that it is designed to work for !!! [You can tell what I really think of it.] But there is a very deep mathematical question hidden behind the technique that most folk seem not to realise. Can you deal with the question I pose?



To factor a polynomial of the form ax^2 + bx + c
Example, 3x^2 + 11x + 10
Let's call (a * c) the "Master" product, a * c = 3 * 10 = 30
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, 30
The prime factors are: 2 * 3 * 5
Find two factors of a*c that add to the middle term 'b'
(coefficient of x)
b = 11
only (2 * 3) + 5 = 11
So we write, 3x^2 + 6x + 5x + 10
Group the terms to form pairs, 3x(x + 2) + 5(x +2)
Factor out the shared (common) binomial parentheses
(3x + 5)(x + 2)

Why does this technique work?