First of all you should always remember that if you arrive at some expression you can't solve it doesn't mean that it's not correct, get yourself free from this limiting way of thinking which is the result of improper teaching methods.

I've rechecked your solution, and your characteristic polynomial is correct.

Now if only you could have one root of this polynomial you could reduce it to a second degree polynomial by means of long division which you could readily solve. Fortunately there's a simple method to find this root:

Rational root theorem - Wikipedia, the free encyclopedia

to assure you that this method in fact works try substituting 8 into your equation.