The answer is x²-y² + xy²+x²y/x-y
Solution:
By polynomial grid division, we start by the divisor (x - y) placed on the row headings of the table and end with the quotient on the column headings.
We know that x² must be in the top left of the grid so that the row and column multiply to x³. We multiply x² by the terms of the row headings to fill in all of the first column:
x²
x x³
-y -x²y
We got -x²y though we want y³. The next quadratic entry must then be -y² to get y³. Multiplying -y² by the divisor, we fill in all of the second column:
x² -y²
x x³ -xy²
-y -x²y y³
We end up with a grid sum -x²y - xy² which tells us that we have a remainder of xy² + x²y that we write next to the grid:
x² -y²
x x³ -xy² remainder: + x²y + xy²
-y -x²y y³
We have to add the remaining fractional part to the quotient that we can read off the first row. Therefore,
x³+y³ / x-y = x²-y² + xy²+x²y/x-y
