Apply the Rational Roots Test to obtain a list of possible (potential) rational (whole-number or fractional) roots.
Use synthetic division to check each of these potential zeroes. If any works, then continue working from the remaining (smaller) polynomial.
Once you get down to a quadratic, apply the Quadratic Formula to find the last two solutions.
Note that you can use a graph to get a quick fix on the best of the potential rational roots to try. And keep in mind that any place where the graph touches the x-axis, but does not cross, indicates a repeated root. So if a rational root, such as x = 1, works, and the graph shows the line not actually crossing at x = 1, try the same root in the synthetic division again.
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