Sum of integral values of n such that  $\sin x (2 \sin x + \cos x) = n$ has at least one real solution is 

$\begin{array}{l}2{\sin }^{2}x+\frac{2\sin x\cos x}{2}=n\\ \Rightarrow \sin 2x+2(1-\cos 2x)=2n\\ \Rightarrow \sin 2x-2\cos 2x=2(n-1)\\ \Rightarrow -\sqrt{5}\le 2n-2\le \sqrt{5}\\ \Rightarrow 1-\frac{\sqrt{5}}{2}\le n\le 1+\frac{\sqrt{5}}{2}\end{array}$

Integral values of $n$ are $0, 1, 2.$