Explanation

[Solution by Dr. Abubacker] Let $t=x^2-9x+11$; then $(x-4)(x-5)=t+9$ and the equation becomes $t^2-t-12=0$, so $t=4$ or $t=-3$. $t=4$: $x^2-9x+7=0$ has irrational roots. $t=-3$: $x^2-9x+14=0$ gives $x=2,7$. The product of rational roots is $14$. Answer (1).