There are a few different approaches. Off the top of my head...

# assume $_ holds the value I'm testing

# 1: Two tests
if ( /^(?!cat)/ && /dog$/ ) {} # if does not start with cat && ends wi
+th dog

# 2: take advantage of zero-width assertion in (?!cat)
if ( /^(?!cat).*dog$/ ) {} # if we start with something that is not 'c
+at'
                           # then have 0 or more chars before ending w
+ith 'dog'

# 3: not using regex at all!
if ( 
   index($_, 'cat') != 0 
   && index($_,'dog') == length($_) - 3 
) {}  # if we don't find 'cat' at the head of the string, 
      # && find 'dog' at the end.
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