The basic idea is to guess an x, put it in the right hand side, calculate the function. We'll do the example calculation and tabulate the results. We could at this point presume that this is really a root, try synthetic division, and factor the polynomialThis method can be used with many, but not all, polynomials. Numerical methods can work very well, like the example above. In a wide variety of cases, this method is very frustrating because the right-hand side of the equation bounces around and does not converge nicely.
Numerical analysis is a big topic, and gets technical very fast.
See the other examples on numerical methods for further tips. This gives the maximum number of real roots. Use the rational zeros theorem to guess possible rational roots. Try each one quickly via synthetic division. If one or more turns out to really be a root, factor the polynomial as much as possible.