The possible rational roots are of the form plus-or-minus p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
You can make the list as easily as we can. p is either 1 or 5; q is either 1, 2, 3, or 6.
You can go through the list of possible rational roots and try each one using synthetic division; but that is tedious and not a good use of your time.
One way to start on the problem and make much better use of your time (still giving you practice with synthetic division) is to use a graphing calculator to find one of the roots and go from there.
Or you can do some logical reasoning, using Vieta's Theorem to see that the sum of the roots is -25/6 and the product of the roots is -5/6.
That suggests that -5 is one of the roots; and synthetic division confirms that:
-5 | 6 25 -24 5
| -30 25 -5
+-----------------
6 -5 1 0
So
Then the quadratic factors nicely:
And so the roots are -5, 1/3, and 1/2.