Overall system IR: How to do arithmetic with $\delta$ functions?
You make it unnecessarily complicated.

Given 
$$\begin{align}
h_1[n]&amp;=2\delta[n-2]-3\delta[n+1]\\
h_2[n]&amp;=\delta[n-1]+2 \delta[n+2]
\end{align}$$
to calculate
$$h[n]=h_1[n]\star h_2[n]$$

the easiest way is to use the sampling property of delta, which is 


  $$f[n]\star\delta(n\pm n_0)=f[n\pm n_0]$$ 


So you can write as follows

$$\begin{align}
h_1[n]\star h_2[n] &amp;= h_1[n]\star \left(\delta[n-1]+2 \delta[n+2]\right)\\
&amp;=\color{green}{h_1[n-1]}+2\color{blue}{h_1[n+2]}\\
&amp;=\color{green}{2\delta[(n-1)-2]-3\delta[(n-1)+1]}+2\color{blue}{\left(2\delta[(n+2)-2]-3\delta[(n+2)+1]\right)}\\
&amp;=2\delta[n-3]-3\delta[n]+4\delta[n]-6\delta[n+3]
\end{align}$$