There must be a typos somewhere if they expect you to find rational zeros. There are none.
Or, they knew that, and just want you to use some numeric methods like bisection, secants, Newton's method, etc.
However, since the problem is in a section discussing the Factor Theorem, I'd expect there to be at least one rational root, so check for a typo.
Use the Factor Theorem to determine the rational zeros of this function. Then use other methods to determine any other zero of this function.
4x^4-x^3-11x^2-2x+6
(I already did p/q and found a total of 16 possible zeros. I synthetic divide all of them and none gave me a root of zero. What do I do next?)
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