After multiplying by (2x) (x+1) x(3x+3) and simplifying I get 3x^3+12x^2+9x=2x^2+2x. This is the part where I'm sort of lost; I'm not sure if I made a mistake or not.
No mistakes so far. Notice that every single term has an x in it. Since x = 0 is not a solution of the original equation (you'd divide by zero in the first term!), you can cancel a power of x from every term. What do you suppose you could do next?