Q:

division of polynomials of same degree

I am solving a differential equation which requires solving and integrating $(4y^3 + 10y^2 + 5y + 2) / (2y^3 + 3y^2 + 3y + 1)$. I am thinking synthetic division to reduce it but am not sure where to start as both polynomials are of the same degree. 

A:

When you divide polynomials of the same degree the quotient is a constant and the remainder is usually of one degree less.  The quotient comes from the ratio of the leading terms.  
Here you have $$\frac{4y^3 + 10y^2 + 5y + 2 }{ 2y^3 + 3y^2 + 3y + 1}=2+\frac{ 4y^2 -y  }{ 2y^3 + 3y^2 + 3y + 1}$$

