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Solution
$$\sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} =$$ =Â Â $$\sqrt{2^2+\left(-\sqrt{\ 5}\right)^2+\left(2\sqrt{\ 3}\right)^2+2\cdot2\cdot-\sqrt{5}+2\cdot\left(-\sqrt{\ 5}\right)\left(2\sqrt{3}\right)+2\cdot\left(2\sqrt{3}\right)\cdot2}=$$
Which is in the form $$a^2+b^2+c^2+2ab+2bc+2ca\ =\ \left(a+b+c\right)^2$$
hence $$\sqrt{21 - 4\sqrt{5} + 8\sqrt{3} - 4\sqrt{15}} =$$ = $$2-\sqrt{\ 5}+2\sqrt{\ 3}$$
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