3x² - 14x - 5

Your error is that you don't want two numbers that will give you
a SUM of 14 and a product of 15.  

Instead you want two numbers that will give you a DIFFERENCE of 14 
and a product of 15.

The way you tell whether you want a SUM or a DIFFERENCE is by
looking at the last sign in 3x²-14x-5, which is the sign of -5,
which is NEGATIVE, so that tells you that you want a DIFFERENCE
of 14, not a SUM of 14.

The factors of 15 are

15×1 and the difference is 15-1 which is the 14 you want.
5×3  and the difference is 5-3 which is 2 which you DON'T want.

So use 15 and 1 to rewrite the middle term -14x by giving them
the proper signs.  The larger is 15, and it gets the same sign
as the middle term of 3x²-14x-5, which is -,  so here goes:

3x²-14x-5

3x² - 15x + 1x - 5

Factor 3 out of the first two terms and +1 out of the last three:

3x(x - 5) + 1(x - 5)

(x - 5)(3x + 1)

I have a tutorial on the AC method of factoring.  Go here to see it:

http://www.algebra.com/my/change_this_name32371.lesson?content_action=show_dev

Edwin