Factorising question help

I'm confused about this question: Factorise 3x² -19x+20
I started to do my usual method for Factorising when the a term isn't 1:
1) C term times A term= 3x20=60
2) Find factor pairs of this new number: for 60 this would be 1 and 60, 2 and 30, 3 and 20, 5 and 12, 6 and 10.
3) This would be to see what pair adds to get the b term and work out which (if any) needs to be negative. This is where I am stumped. none of these pairs add to make 19.
Does this work? If it does, can someone tell me what I'm doing wrong?
Thanks!

Youve missed a factor pair 15,4

that would explain why it wasn't working 😅.
Thank you.

Note that if all else faills you can always use the quadratic formula to get the roots and reverse engineer the factors from them.

Also note that when you list the factor pairs
1+60=61
2+30=32
3+20=23
5+12=17
...
You know they both have the same sign (negative, as the product is postive) so their sum must lie between 17 and 23, so there must be a factor pair beween 3,20 and 5,12 ...

I don't understand this.

If you chucked the quadratic=0 into the formula to get the roots youd get
x = (19 +/- 11) / 6 = 5, 4/3
so which corresponds to a factorisation
(x-5)(x-4/3) = 0
and noting the original quadratic was 3x^2 ... so multiply through by 3 gives
(x-5)(3x-4)
which is what you want.

Its laborious compared to spotting the factors, but its "turn the handle".