There are lots of ways to do this but perhaps one of the most straight forward ways is to simply make a table of values.
The intermediate value theorem says that a continuous function (such as the polynomial in this problem) obtains all values between it's extrema in a given interval. In other words if at one point y is negative and at another y is positive than somewhere in between it had to be 0.
We look to our list and see that x=-3, f(x)=42 and at x=-2, f(x)=-7. At x=-3, f(x) is positive and at x=-2, f(x) is negative, so somewhere between x=-3 and x=-2 f(x) had to be 0, by the intermediate value theorem. Thus there is a zero somewhere in the interval (-3,-2). Can you find another interval using the same method?