Poles and zeros of a transfer function What are the poles and zeros of this transfer function (in $z$):

$$H(z)=z+2+z^{-1}$$

and how would you approach the resolution of such problem?

Personally, I would write 
$$H(z)=\displaystyle\frac{z^2+2z+1}{z}=\frac{(z+1)^2}{z}$$ then I would say that there is a zero (double) at $z=-1$ (numerator $= 0$) and a pole at $z=0$ (denominator $= 0$). 


But wouldn't we also have a pole at $z=\infty$ since it makes $H(z)\to \infty$ which corresponds to the definition of a pole? 
And in that case wouldn't that be contradictory since we know that $H(z)$ can be seen as a transfer function of a FIR filter which we know is always stable, so how can the filter be stable if it has a pole at $z=\infty$ (outside unit circle)?


I'm sure I have a basic misunderstanding about poles and zeros otherwise there shouldn't be any contradiction and hopefully someone can help me clarify this :-)
The "poles-inside-unit-circle" stability criterion only applies to causal systems. Your system is not causal because it uses one sample from the future owing to the $z$ term.

The general technique to check for stability involves looking at the regions of convergence (ROC) of $H(z)$. If the ROC includes the unit circle, then the system is stable.

See also this answer to the question: What is the relationship between poles and system stability?