Here is the sentence again:

Derek wanted a ferocious pit bull terrier to scare off burglars, but his apartment complex did not allow dogs as a result, Derek had to settle for naming his goldfish Mr. Jaws and installing additional door locks.

His apartment complex did not allow dogs is a main clause. As a result, Derek had to settle for naming his goldfish Mr. Jaws ... is a second main clause. Because these two clauses run right into each other with no break, they create a fused sentence.

A semicolon would go nicely in the empty space between dogs and as.

Go to the next sentence.

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