Question 6
Do there exist polynomials $p(x),q(x),r(x)$ whose coefficients are positive integers, satisfying
$$p(x)=(x^2-3x+3)q(x),\qquad q(x)=\left(\frac{x^2}{20}-\frac{x}{15}+\frac1{12}\right)r(x)?$$
$$p(x)=(x^2-3x+3)q(x),\qquad q(x)=\left(\frac{x^2}{20}-\frac{x}{15}+\frac1{12}\right)r(x)?$$