Polynomial Diophantine Equation 
If $x$,$y$ $\in \mathbb Z$, find all the solutions of
$$y^3=x^3+8x^2-6x+8$$

I have tried factorizing the equation but the polynomial on $\text{R.H.S.}$ doesn't have any integral roots. Further, I deduce that the parity of $x$ and $y$ is same. However, I can't seem to find a way to go further.
Any help will be appreciated.
Thanks!

In general for these type of equations there are good upper bounds for $|x|.$ For instance see theorem 4 of a paper of G.Walsh [http://matwbn.icm.edu.pl/ksiazki/aa/aa62/aa6225.pdf].
This theorem gives $|x|\leq 3^9\cdot 9^6=3^{21}.$ Then you can check (having a little patience) all such $x$ (you need $2^{33}$ rounds and check only for $|x|<0$ due to the answer of the previous poster; it is doable).
