With roots -1 and 3, two of the linear factors of the polynomial are (x+1) and (x-3).
Use synthetic division to factor out one of those factors, leaving a polynomial of degree 3.
Then see if that remaining polynomial can be factored by any method you know; if not, then use synthetic division again to factor out the other of the known factors, leaving a quadratic polynomial.
Then, if the remaining roots are rational, the quadratic will be factorable to find two other rational roots; if not, you will need to use the quadratic formula.
In case you don't know synthetic division, here is how to factor the linear factor (x-3) out of the given polynomial.
3 | 1 -2 -2 -2 -3
| 3 3 3 3
+---------------------
1 1 1 1 0
The reduced polynomial is
You might see that this polynomial can be factored by grouping; then the rest of the problem should be easy.