First step is to factor out the x. So now P(x) = x(x3 - 2x2 - 13x -10)
One method to factor this is to look at the constant, in this case the -10 and think of all of the factors that could multiply to equal it. In this case that would include +/- 1, +/- 2, +/- 5, +/- 10.
From here its a little bit of process of elimination. Try synthetic division or long division, whichever you are more comfortable with, and see if any divide evenly. I started with (x - 1) but it didn't work. I then tried (x + 1) and it divided evenly with an answer of x2 - 3x - 10. Then by observation, we need two numbers to multiply to equal -10 and to add to equal -3. So that then factors into (x - 5)(x - 2).
Therefore, P(x) = x(x + 1)(x - 2)(x - 5)
So the real zeros are -1, 0, 2, 5
Another way to help with the initial guessing of the synthetic or long division is to graph the function on your calculator. The graph of the curve will cross the x-axis at the zeros.
Just don't graph it and assume they go directly through whole numbers. What looks like going through 0 on the graph, depending on your scale, could be .002 or some other close number. Always follow through with factoring it.