The expression $$\sqrt{10+2\left(\sqrt{6}-\sqrt{15}-\sqrt{10}\right)}$$ is equal to:

$$\sqrt{10+2\left(\sqrt{6}-\sqrt{15}-\sqrt{10}\right)}$$
= $$\sqrt{3 + 2 + 5 +2\left(\sqrt{3} \times \sqrt{2} - \sqrt{2} \times \sqrt{5} -\sqrt{5} \times \sqrt{2}\right)}$$
= $$\sqrt{(\sqrt{3})^2 + (\sqrt{2})^2 + (-\sqrt{5})^2 +2\left(\sqrt{3} \times \sqrt{2} - \sqrt{2} \times \sqrt{5} -\sqrt{5} \times \sqrt{2}\right)}$$
= $$\sqrt{(\sqrt{3} + \sqrt{2} -\sqrt{5})^2}$$
= $$\sqrt{3} + \sqrt{2} -\sqrt{5}$$