Because the sum of the powers in the terms are constant, by making y=1 we can use synthetic division using -5/3 as a zero:
-5/3 | 15..19..-7...5
.........15 -25 10..-5
.........15...-6...3 | 0
The quotient is therefore (15x^2-6xy+3y^2)/3=5x^2-2xy+y^2. The division by 3 was necessary because we used -5/3 as the zero corresponding to 3x+5, which was divided by 3 to give x+5y/3. That means the quotient obtained through synthetic division was 3 times too large.