Question 22876
This is what I call "Advanced Trinomial Factoring".  It might NOT be a lesson in algebra.com, but you can find a lengthy explanation of this in my book at my website.  Click on my user name "rapaljer" anywhere in algebra.com.  Then click on "Basic Algebra", then "Samples of Basic Algebra: One Step at a Time",  go to "Chapter 2", and look for "Advanced Trinomial Factoring."


Now, having given that notice for my free website,


15X^2+19X+6=0


It pretty much has to start with 
(5x _____   )(3x _____  ) = 0


Now, look at the last times last, which has to end up with a +6.  That would most likely have to be 3 times 2, or 2 times 3.  Now, try putting the 3 in the first set of parentheses and the 2 in the second set of parentheses.  In my chapter on this topic, I explain how I knew to do that in more detail.  For now it looks like this:

(5x___   3)(3x ___  2) = 0.  


Now do the outer times outer and inner times inner and see if it adds up to 19x.  Outer times outer = 10x; inner times inner = 9x; and sure enough it adds up to 19x.  


(5x+3)(3x+2) = 0


First solution:
5x+3 = 0
5x= -3
{{{x=-3/5}}}


Second solution:
3x+2 = 0
3x= -2
{{{x=-2/3}}}


R^2 at SCC