Question

I have chosen 4 MYSTERY INTEGERS. At least one is positive and one is negative. Follow these clues to figure out my integers. BE SURE TO EXPLAIN YOUR REASONING!
1. The sum of the four integers is (-6)
2. When you order the integers from LEAST to GREATEST the difference of the middle two integers is (-4) and the sum is 0
3. The quotient when you divide the LEAST integer by the GREATEST is (-3)
4. The product of the two greatest integers is (+6)

Answer 1

Let the four numbers be A,B,C,D in ascending order.

The key is in conditions 2 & 3.
B-C=-4, and B+C=0
therefore B=-2, C=2 by solving the two previous equations.

Given that the product of the two greatest numbers is 6, we have C*D=6, or 
D=6/C=6/2=3
Now since the sum of all the four numbers is -6, we have
A+B+C+D=-6
A-2+2+3=-6
A=-6+2-2-3=-9
Check: we were told smallest divided by greatest = A/D=-9/3=-3  ok.

So the numbers are
A=-9, B=-2, C=2, D=3.