Let the numbers be $\eta$ and $\zeta$.
$$\dfrac{\eta+\zeta}{2}=6\Rightarrow \eta+\zeta=12$$.
$$\sqrt{\eta\zeta}=10\Rightarrow \eta\zeta=100$$.
The monic quadratic with roots $\eta$ and $\zeta$ is $x^2-(\eta+\zeta)x+\eta\zeta$. Therefore, an equation with $\eta$ and $\zeta$ as roots is $x^2 - 12x + 100 = 0\Rightarrow \text{(D)}$