i still cant find a proper solution if the question ask number of factor more than 4 but for this question we can check that $2,3,5,7\nmid n^2-n+11$ thus $n^2-n+11 \geq 11^4$ we can manualy check that $n^2-n+11\neq 11^4$ there for $n^2-n+11\geq11^3(13)$ since $n^2-n+11$ is strictly increasing $\forall n \in\mathbb{N}$ $\Rightarrow$ the smallest n will occur at $n^2-n+11=11^3(13)$ which is $n=132~~\square$