Factorise:

12x² + 17xy - 5y²

Multiply the 12 by 5, disregarding the sign, getting 60.

Write down a list of all the ways to factorise 60 using two 
positive integers:

60*1 = 60
30*2 = 60
20*3 = 60
15*4 = 60
12*5 = 60
10*6 = 60

Now look at the sign of the last term to see whether we add or
subtract those pairs of numbers.  The last term is -5y², which
has a - sign, so we subtract those same pairs of numbers that
we multiplied above to get 60. 
(when the last term has a + sign, we add, but since it has a -, we subtract):

60-1 = 59
30-2 = 28
20-3 = 17
15-4 = 11
12-5 =  7
10-6 =  4

And we look down that list to find the coefficient of
the middle term of 12x² + 17xy - 5y², in absolute value,
which is 17 and we find it as

20-3 = 17

So we substitute (20-3) in place of 17 

12x² + 17xy - 5y²
12x² + (20-3)xy - 5y²

and remove the parentheses:

12x² + 20xy - 3xy - 5y²

Next we factorise only the FIRST two terms by
factorising out 4x:

4x(3x + 5y) - 3xy - 5y²

Next we factorise only the LAST two terms by
factorising out -y:

4x(3x + 5y) - y(3x + 5y)

Notice that the two parentheses are the same.
I'll colour them red:

4x(3x + 5y) - y(3x + 5y)

So factorise out the red factor, leaving the black terms in parentheses:

(3x + 5y)(4x - y)

Edwin