You can solve this equation in closed form, as discussed in other answers, but to be honest, closed-form solutions to polynomials of degree > 2 are not very useful in practice, because the results tend to be poorly conditioned.
For your particular polynomial, I agree with Alexandre that Newton's method is probably the way to go.
In the long run, though, I highly recommend writing (or reusing from the Internet) an implementation of the Jenkins-Traub root-finding algorithm. Wikipedia describes it as "practically a standard in black-box polynomial root-finders," and they're not exaggerating. It has served all of my polynomial-solving needs for years; in my experience it's more robust than Newton's method (no reliance on a good initial guess) and eigenvalue-based methods, and is quite fast to boot.