You look for rational zeroes. You only need to find one to solve the problem.
Take the factors of the leading power"s coefficient: 5 has factors 1 and 5.
Take the factors of the lowest power's coefficient (the constant): 42 has factors 1, 42, 2, 21, 3, 14, 6, 7.
The rational zeroes are 1, 42, 2, 21, 3, 14, 6, 7, 1/5, 42/5, 2/5, 21/5, 3/5, 14/5, 6/5, 7/5 in positive and negative form. Substitute each one for x until you get zero in the function.
x=1 gives us: 5-12-23+42=12 (not zero): x=-1 gives us: -5-12+23+42=48 (not zero).
x=2 gives us 40-48-46+42=-12; x=-2 gives us -40-48+46+42=0 (bingo!). x=-2 is a zero. Divide by this zero using synthetic division:
-2 | 5 -12 -23 42
5 -10 44 | -42
5 -22 21 | 0 = 5x2-22x+21 = (5x-7)(x-3).
So we have all the zeroes: -2, 7/5, 3.