2001 AMC 12 Problem 23

Writing $P(x)$ $= (x - r)(x - s)(x^2 + \alpha x + \beta)$ with integer roots $r, s,$ matching coefficients forces $\alpha$ and $\beta$ to be integers. The other two zeros are $$-\dfrac{\alpha}{2} \pm \dfrac{i\sqrt{4\beta - \alpha^2}}{2}.$$ A real part of $\dfrac{1}{2}$ requires $\alpha = -1,$ making the imaginary part $\dfrac{\sqrt{4\beta - 1}}{2}.$ Choice A needs $\sqrt{4\beta - 1} = \sqrt{11},$ i.e. $\beta = 3,$ an integer, so it works. The other choices force a non-integer $\beta$ (for example choice E needs $\beta = 3.5,$ and choice D needs $\alpha = -2$ with $\beta = \tfrac{5}{4}$). Thus, the correct answer is A.