How can we use z transform in finding convolution?

Example: Given $x(n)= \delta(n)+ 2\cdot \delta(n-1)- \delta(n-3)$ and $h(n)=2\cdot \delta(n+1) + 2\cdot \delta(n-1)$

z transforms of $x(n)$ and $h(n)$ are $X(Z)=  1 + 2\cdot z^{-1} - z^{-3}$
                        $H(Z) =    2 \cdot z^{1} + 2 \cdot z^{-1}$

convolution in time domain leads to multiplication in z transform
$x(n) \ast h(n) = X(Z) \cdot Y(Z)$

$Y(Z)= X(Z) \cdot H(Z)= 2\cdot z + 4+ 2\cdot z^{-1}+2\cdot z^{-2} -2\cdot z^{-4} $

taking inverse z transform $y(n)= 2\delta(n+1) + 4 \delta + 2 \delta(n-1) + 4 \delta(n-2)-2\delta(n-4)$