Finding inverse z transform for two sided ROC?
Since you have two poles, you have to decompose $0.3 &lt;|z|&lt;1$ into two ROCs, $R_A$ and $R_B$ such that: $$R_A \cap R_B \subseteq \{0.3 &lt;|z| &lt; 1\}$$

Your poles are $z=-0.3$, $z=-1$, so you have two choices for each ROC:


$|z| &gt; 1$
$|z| &lt; 1$


and


$|z| &gt; 0.3$
$|z| &lt; 0.3$


The requested ROC is given by $R_A \cap R_B = \{|z| &lt; 1\}\cap\{|z| &gt; 0.3\} = \{0.3 &lt; |z| &lt; 1\}$

In this case (and using your table),

$$y[]=-2.86(−1)^{}[--1]−1.86(−0.3)^{}[]$$