$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response.

I transformed this system via using Z transform:

Y(Z)    z^4 - 10z^3 - 4z^2 + 4z
---- = --------------------------
X(Z)         2( z^2 -z -2 )

$$H(Z) = 0.5z^2 -4z + \frac{-5}{z-2} + \frac{-1/3}{z-2} + \frac{1/3}{z+1}$$

How to find ROC to inverse the Z transfom to $h[n]$ ?

$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response.

I transformed this system via using Z transform:

$$\frac{Y(z)}{X(z)}=H(z)=\frac{z^4 - 10z^3 - 4z^2 + 4z}{2( z^2 -z -2 )}$$

$$H(z) = 0.5z^2 -4z + \frac{-5}{z-2} + \frac{-1/3}{z-2} + \frac{1/3}{z+1}$$

How to find ROC to inverse the Z transfom to $h[n]$ ?

2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3] is the system that I'm looking for the response.

I transformed this system via using Z transform:

Y(Z)    z^4 - 10z^3 - 4z^2 + 4z
---- = --------------------------
X(Z)         2( z^2 -z -2 )

H(Z) = 0.5z^2 -4z -5(1/(z-2)) -1/3(1/(z-2)) +1/3(1/(z+1))

How to find ROC to inverse the Z transfom to h[n]?

$2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3]$ is the system that I'm looking for the response.

I transformed this system via using Z transform:

Y(Z)    z^4 - 10z^3 - 4z^2 + 4z
---- = --------------------------
X(Z)         2( z^2 -z -2 )

$$H(Z) = 0.5z^2 -4z + \frac{-5}{z-2} + \frac{-1/3}{z-2} + \frac{1/3}{z+1}$$

How to find ROC to inverse the Z transfom to $h[n]$ ?

2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3] is the system that I'm looking for the response.

I transformed this system via using Z transform:

Y(Z)    z^4 - 10z^3 - 4z^2 + 4z
---- = --------------------------
X(Z)    2( z^2 -z -2 )

H(Z) = 0.5z^2 -4z -5(1/(z-2)) -1/3(1/(z-2)) +1/3(1/(z+1))

How to find ROC to inverse the Z transfom to h[n]?

2y[n-2]-2y[n-3]-4y[n-4]=x[n]-10x[n-1]-4x[n-2] + 4x[n-3] is the system that I'm looking for the response.

I transformed this system via using Z transform:

Y(Z)    z^4 - 10z^3 - 4z^2 + 4z
---- = --------------------------
X(Z)         2( z^2 -z -2 )

H(Z) = 0.5z^2 -4z -5(1/(z-2)) -1/3(1/(z-2)) +1/3(1/(z+1))

How to find ROC to inverse the Z transfom to h[n]?