#Python 3, 279 277 258 251 bytes

k=str.replace def f(x): z=len(x) y='--'*(['-1']==[c for c in x if'0'!=c][:1]) for i,v in enumerate(x): p=str(z+~i) if v in'-1'and~i+z:y+='+x^'+p elif'0'!=v:y+='+'+v+'x^'+p return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

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Special thanks to ovs and NK1406.

Python 3, 279 277 258 251 bytes

k=str.replace def f(x): z=len(x) y='--'*(['-1']==[c for c in x if'0'!=c][:1]) for i,v in enumerate(x): p=str(z+~i) if v in'-1'and~i+z:y+='+x^'+p elif'0'!=v:y+='+'+v+'x^'+p return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

Try It Online!

Special thanks to ovs and NK1406.

#Python 3, 279 277 258 bytes

k=str.replace def f(x): y='--'*(['-1']==[c for c in x if'0'!=c][:1]) for i,v in enumerate(x): if v in'-1'and~i+len(x):y+='+x^'+str(len(x)+~i) elif'0'!=v:y+='+'+v+'x^'+str(len(x)+~i) return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

Try It Online!

Special thanks to ovs.

#Python 3, 279 277 258 251 bytes

k=str.replace def f(x): z=len(x) y='--'*(['-1']==[c for c in x if'0'!=c][:1]) for i,v in enumerate(x): p=str(z+~i) if v in'-1'and~i+z:y+='+x^'+p elif'0'!=v:y+='+'+v+'x^'+p return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

Try It Online!

Special thanks to ovs and NK1406.

#Python 3, 279 277 bytes

k=str.replace def f(x): y=''  for i in x:  if'0'!=i if'-1'==i: y='--' break for i,v in enumerate(x): if v in'-1'and~i+len(x):y+='+x^'+str(len(x)+~i) elif'0'!=v:y+='+'+v+'x^'+str(len(x)+~i) return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

Try It Online!

Special thanks to ovs.

#Python 3, 279 277 258 bytes

k=str.replace def f(x): y='--'*(['-1']==[c for c in x if'0'!=c][:1]) for i,v in enumerate(x): if v in'-1'and~i+len(x):y+='+x^'+str(len(x)+~i) elif'0'!=v:y+='+'+v+'x^'+str(len(x)+~i) return y and k(k(k(k(y[1:],'+-','-'),'^1',''),'x^0',''),'-+','-')or 0 

Takes input as a list of strings. This solution is not highly golfed yet. This basically works by replacing things to suit the output format, which highly increases the byte count.

Try It Online!

Special thanks to ovs.