I know that the square root of a square is the absolute value of the original number, so I can rewrite the equation as $|x+3| = 7$. This means that either $x+3 = 7$ or $x+3 = -7$, since both values have the same absolute value. Solving for $x$ in both cases, I get $x = 4$ or $x = -10$. To find the sum of all values of $x$ that satisfy the equation, I just add them up: $4 + (-10) = -6$. 
What is the sum of all values of $x$ for which $\sqrt{(x+3)^{2}} = 7$?