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You can use the rational roots theorem to guess and check the possible roots. You will find that (x + 2) and (x + 6) are factors of this polynomial. The other factor is (x^2 - 5). Can you obtain the rational roots from here?
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Using the Rational Roots Theorem (http://www.mathwords.com/r/rational_root_theorem.htm) you guess and check the possible values. The possible rational roots are +- p/q where p are the factors of 60 and q are the factors of 1. I.e., the possible rational roots are +- 1, +-2, +-4, +-5, +-6., etc... By guess and check we can try x = 1, x = -1, x = -2, x = 2 and x = -2. We find that x = -2 results in a 0. That means (x + 2) is a factor. Divide the original polynomial by (x + 2) to obtain a simpler polynomial of degree 3 and repeat the same steps to obtain the next factor.
I repeated x = -2 twice by accident in the explanation above.
Anyway, to show x = -2 results in a 0:
16 - 64 + 28 + 80 - 60 = 0
So if x = -2, that means (x + 2) must be a factor of the original polynomial. Division yields (see attached file).
Our new polynomial is x^3 + 6x^2 - 5x - 30. We can use the same Rational Roots Theorem to determine the remaining possible rational root(s).