Use synthetic division to find the quotient and remainder:
I know how to do synthetic division with just x, but this threw me for a loop. Bagatrix (math software) also could not solve it. Any ideas?
My guess (and it's only a guess!) is that you are expected to use the underlying concepts for synthetic division, and apply them here.
When you do synthetic division of a polynomial in x by a factor in x, you are actually just plugging in the corresponding x-value. For instance, if you divide by x - 3, you are actually only plugging the 3 into the division and, if the remainder is zero, then you will have shown that x = 3 is a zero of the original polynomial (or, which is the same thing, that x - 3 is a factor).
In this case, you are dividing by x - y. When you were dividing by x - 3, you set this equal to zero, and solved for the value to plug into the synthetic-division matrix: x - 3 = 0, so x = 3, so plug in "3". The corresponding process here would be: x - y = 0, so x - y, so plug in "y".
You can find "the answer" by long polynomial division. Try the synthetic division with the top row being 1, 0y, 0y^2, 0y^3, 1y^4, and see if you get the right result.