Given $n$ is rational, $\sqrt{14}$ is irrational.

$-4\sqrt{14}$ is irrational (rational $\times$ irrational = irrational).

$-4\sqrt{14} - n$ is irrational (irrational $-$ rational = irrational).

$\frac{-4\sqrt{14} - n}{2} \text{ is irrational}$

(An irrational number divided by a nonzero rational is still irrational.)