The answer is not 38/53

Hi Lightning  

$P(x)=ax^2+bx+c\\ P(4-\sqrt{11})=a(4-\sqrt{11})^2+b(4-\sqrt{11})+c=0\\\qquad \text{Where a,b and c are integers}\\ a(4-\sqrt{11})^2+b(4-\sqrt{11})+c=0\\ a(16+11-8\sqrt{11})+b(4-\sqrt{11})+c=0\\ a(27-8\sqrt{11})+b(4-\sqrt{11})+c=0\\ 27a+4b=-c\quad(1)\\ -8a\sqrt{11}-b\sqrt{11}=0\\ -8a-b=0\\ b=-8a \quad (2)\\ \text{sub two into one}\\ 27a+4*-8a=-c\\ -5a=-c\\ c=5a$

So the equation becomes

$P(x)=ax^2-8ax+5a\\ P(x)=a(x^2-8x+5)\\ $

Find P(3) and P(4) and do the division.

You can finish it   

(oh, you do need to check my working, I have not done so. )