ROC and impulse response
It seems that you've correctly found the two poles: $z_{\infty,0}=2$ and $z_{\infty,1}=-\frac13$. The only other thing you need to know is that the possible ROCs are limited by the pole radii, and that there can be no poles inside an ROC.

Note that $z&lt;2$ doesn't make much sense because $z$ is a complex number. If you mean $|z|&lt;2$ then the pole at $z=-\frac13$ would be inside that ROC, so $|z|&lt;2$ is not a valid ROC for the given transfer function.

The three possible ROCs for your example have the following forms:


$|z|&lt;a$
$a&lt;|z|&lt;b$
$|z|&gt;b$


The numbers $a$ and $b$ are of course related to the poles of the transfer function. I trust that you can take it from here.