$q\left(x\right)=x^4+x^2$ , $r\left(x\right)=x+1$ ,

$y\left(x\right)=x^6+x^2+1$ , $y_2\left(x\right)=x^2+1$ , $y_1\left(x\right)=x^6+x^2+1$ ,

$P\left(x\right)=x^2$ , $A\left(x\right)=x+1$ .