I started by rearranging the equation into (29x-3)^2013 + 29x - 1 = 0.
Then I substituted y=29x-3 to get f(y)=y^2013+y+2=0. There is one obvious real solution y=-1.
The first derivative f'(y)=2013*y^2012+1 is always greater than zero which means f(y) is continuously increasing and then must have only the one real root y=-1.
Solution