In order to answer this question correctly, we must first understand how to punctuate dialogue.
Dialogue is always set off by quotation marks. A comma should appear after the word "said." Consider this example: Bobby said, "This is my house." Note that a comma appears after the word "said." Quotation marks are used to set off the words spoken by Bobby. Finally, a period appears at the end of the sentence (which is also the end of the sentence).
Now, let's look at the above example: My mother promised, "As long as you complete your homework; you can go to the game."
A comma correctly appears after the word "promised" (which is acting like the word "said"). Furthermore, quotation marks correctly set off the dialogue, and a period appears at the end of the sentence. All looks well.
But what about that semicolon? A semicolon is used to separate two complete sentences (or two independent clauses). Now, consider: are the two halves of dialogue complete sentences?
"You can go to the game" is a complete thought. On the other hand...
"As long as you complete your homework" is NOT a complete thought and could not stand alone as a sentence. As a result, the semicolon is not correct here.
The proper punctuation for this sentence, then, would be:
My mother promised, "As long as you complete your homework, you can go to the game."
Therefore, the correct answer to this question is B.