From the following example in the book

$\displaystyle 24x^2 - 34x - 45$

Step 1

Factors of 24

1 x 24

2 x 12

4 x 6

3 x 8

Step 2

Factors of 45

1 x 45

5 x 9

3 x 15

Look at the sign of CIn this case C is negative, therefore find a value from step 1 list and another from step 2 list such that the difference of their product and the product of two remaining numbers from these steps results in b.

Choose a product from Step 1 & 24 x 6 (e, f) form "a" and 5 x 9 (g, h) form "c"

Here is where I am having difficulty understanding

here is where I am heavily confused, on more than one occasion the author seems to multiply the above in any combination to get the correct product. Is there any method to this madness??

Problem 1) They solve by using the product of 4 and 5 = 20. And the product of 6 and 9 = 54. 54 - 20 = 34, or our b value in this expression.

Problem 2) Then the author pairs up the binomials

(4x - 9) (6x + 5). Here the author says the binomials must be ordered as (e h) (f g). That is fine, but for the previous step I do not understand why they are mixed up.

Many thanks for any help