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Choice D is correct. Adding $10$ to each side of the given equation yields $x^2-40x=10$. To complete the square, adding ${\left(\frac{40}{2}\right)}^2$, or ${20}^2$, to each side of this equation yields $x^2-40x+{20}^2=10+{20}^2$, or ${\left(x-20\right)}^2=410$. Taking the square root of each side of this equation yields $x-20=\pm \sqrt{410}$. Adding $20$ to each side of this equation yields $x=20\pm \sqrt{410}$. Therefore, the solutions to the given equation are $x=20+\sqrt{410}$ and $x=20-\sqrt{410}$. The sum of these solutions is $\left(20+\sqrt{410}\right)+\left(20-\sqrt{410}\right)$, or $40$.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.