Given the input and the output, how to determine the impulse response? I would like to find the impulse response, $h[n]$, of an LTI system given the input

$$x[n] = [1,-3,2]$$

and the output

$$y[n] = [1,-1,-4,4]$$

I know that $y[t]=x[t]*h[t]$, but I am having hard time to figure out the right way to calculate the impulse response. I know very little about signal processing, so if you don't mind giving an easy explanation, then I appreciate it. Or, if it's possible to do an example, that's better.
$\mathcal Z$-transforming the input and the output and then dividing them, we obtain the transfer function

$$\begin{array}{rl} H (z) &amp;= \dfrac{1 - z^{-1} - 4 z^{-2} + 4 z^{-3}}{1 - 3 z^{-1} + 2 z^{-2}}\\\\ &amp;= \dfrac{z^3 - z^2 - 4 z + 4}{z (z^2 - 3 z + 2)}\\\\ &amp;= \dfrac{(z-1)  (z-2) (z+2)}{z (z-1) (z-2)}\\\\ &amp;= \dfrac{z+2}{z} = \color{blue}{1 + 2 z^{-1}}\end{array}$$

Computing the inverse $\mathcal Z$-transform of $H (z)$, we finally obtain the impulse response.