Determine if system is linear time variant
To be time invariant you would have to show that:
$$y(n-k) = (n-1)x(n-k-1) + (n+1)x(n-k+1)$$
(You shift the $x$ and you get a shift in $y$).
But you can't get rid of the $(n-k-1)$ and $(n-k+1)$.
In general any time you have the time term $n$ by itself it will be time variant unless it cancels out somehow.