Why does the general method Factorising quadratics, work?

We are trying to factorise a general quadratic ax²+bx+c [1] into the form (dx+e)(fx+g). Multiplying this out gives:

dx(fx+g)+e(fx+g) [2]

dfx²+dgx+efx+eg [3]

dfx²+(dg+ef)x+eg [4]

If a = df, b = dg+ef and c = eg, then equation [1] is same as equation [4].

So ac can be written as dfeg = dgef = (dg)(ef).

The method requires 2 numbers which when multiplied together make ac = dg ef and when added together make b = dg+ef. Once we have these 2 numbers (dg and ef), we can guarantee that the expression can be fully factorised by factorising dg and ef into d, g, e and f. (Even if the 2 numbers dg and ef don’t have any factors, they will always have factors of themselves and 1).