SOLUTION: (x+5)/(-x^2) How would you do syntheic division here?

There is no way, for the degree of the numerator is less than
the degree of the denominator.  Long or synthetic division can
only be done if the degree of the numerator is NOT less than
the degree of the denominator. 

However, if it had been  instead, then there
would be a way to perform synthetic division.  You would then
be dividing -x² by x+5.  And first you would need to write:

 as .

Then we would set x+5=0 and get x=-5 and use -5 on
the left of the synthetic division:

-5|-1  0   0
  |    5 -25
   -1  5 -25

And the answer would be  

But divided the way you have it stated, synthetic division is
impossible.

Edwin