Rather than instruct you to guess, it would have been better for them to tell you about the Rational Roots Theorem.
Are you familiar with it?
If a polynomial has integer coefficients, then any rational roots must be of the form x = p/q where p is a factor of the constant term and q is a factor of the leading coefficient.
The constant term is 6; its factors are 1, -1, 2, -2, 3, -3, 6, and -6. (These are the possible candidates for p.)
The leading coefficient is 10; its factors are 1, -1, 2, -2, 5, -5, 10, and -10. (These are the possible candidates for q.)
Now, it's just a matter of testing the possibilities until you find one that works.
Have fun. Let us know if you need more help finding rational roots using this theorem. (Hint: there is only one rational root for the polynomial that you posted.)