All right, I get it. It explains why I couldn't follow some steps with physics when they rewrote functions. It seemed complicated but it is quite simple.

$\displaystyle \frac{3x^4+4x^3+6px^2+4qx+r}{x^3+3x^2+9x+3}=x+1+\f rac{(6p-12)x^2+(4q-12)x+r-3}{x^3+3x^2+9x+3}$

The remainder has to be zero because the definition of divisionable is that there has to be no reamainder.

So it turns out that I had two problems with this question:

I hadn't learned how to execute a polynomial division and I didn't know the definition of divisionable. This definition is, for me, counterintuitive. I assumed that every fraction wich is a rational number (domain of $\displaystyle \mathbb{Q}$) would be defined as divisionable.