$$\begin{gathered} \text{Sum of the values of x satisfying the equation } \\ \sqrt{2x+\sqrt{2x+4 }}=4 is \end{gathered}$$

$\begin{array}{l}\text{The given equation is }\\ \sqrt{2x+\sqrt{2x+4}}=4\\ \text{Squaring on both sides}\\ 2x+\sqrt{2x+4}=16\\ \text{}\sqrt{2x+4 } = 16-2x\text{}\\ \text{Squaring on both sides }\\ \text{}2x+4=4{\left(8-x\right)}^2\text{} \\ x+2=2\left(64-16x+x^2\right)\\ \text{}2x^2-33x+126=0\text{}\\ \left(2x-21\right)\left(x-6\right)=0\text{}\\ x=6 or \frac{21}{2}\text{}\\ \text{But }x=\frac{21}{2} \text{does not satisfy the given equation}\\ \text{So\hspace{0.17em}}x=6\end{array}$