What is the algebraic expression represents the phrase “the quotient of negative eight and the sum of a number and three”?

You set it up like this.

So quotient means that you're dividing, so put a ÷
Next, it says the dividend, or the part being divided is -8, so now you have this: -8 ÷
Lastly, it says the divisor, or the part doing the dividing, is a quantity equal to the sum of "some number" (we'll use x) and 3. Since the divisor is one quantity collectively, you have to put it in parentheses.
This gives you: (x + 3)
When you put it all together, you get -8 ÷ (x + 3).
This can also be written as a fraction by putting -8 over (x+3).