I have the diagram above. I found the transfer function below from it;
The question asks me to find out if the system is causal and stable, but didn't it have to indicate whether it was left-sided or right-sided?
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It looks like a Homework problem so I will only provide small hints:
Check from the block diagram implementation of the system whether the current output sample $y[n]$ depends on only current input sample $x[n]$ and past input & output samples $x[n-n_0]$ and $y[n-m_0]$. If so, the system is causal, else it will be non-causal.
You can use the information from the previous point (causal or non-causal) along with the pole location of $H(z)$ to check if the system is BIBO stable or not.
Any Causal System with poles inside unit-circle are BIBO stable.
You can refer to this Pole-Zero Analysis for better understanding.