The error says:

Can't find string terminator '"' anywhere before EOF

and your code has:

print "$last_name;
#     ^
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Nothing to do with the regex.

Actually, the regex would be better not capturing the comma, and there is no need to quote the variable at all:

my $total_names = "Matthew,Thomas,Peter,Randy,George,Federick";
$total_names =~ /,([^,]+)$/;
print $1 if $1;
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or

my $total_names = "Matthew,Thomas,Peter,Randy,George,Federick";
my ($last_name) = $total_names =~ /,([^,]+)$/;
print $last_name if $last_name;
[download]

Update: If your teacher does allow you to use split, you’ll get the last substring by subscripting with -1:

my $last_name = ( split /,/, $total_names ) [ -1 ] ;
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(A [ 0 ] subscript gives you the first substring.)

Hope that helps,

print "$last_name;
      ^----------^
      |          \--missing quote
first quote