Answer :
[tex](a+b)^2=a^2+2ab+b^2\ \ \ (*)\\\\\\\sqrt{4+2\sqrt3}=\sqrt{3+2\sqrt3+1}=\sqrt{\underbrace{(\sqrt3)^2+2\cdot\sqrt3\cdot1+1^2}_{(*)}}\\\\=\sqrt{(\sqrt3+1)^2}=|\sqrt3+1|=\sqrt3+1[/tex]
[tex](a+b)^2=a^2+2\cdot a\cdot b+b^2\\ \\4+2 \sqrt{3}=1^2+2\cdot1\cdot \sqrt{3}+ \sqrt{3} ^2\ \ \ \Rightarrow\ \ \ a=1\ \ \ \wedge\ \ \ b= \sqrt{3} \\ \\4+2 \sqrt{3} =(1+ \sqrt{3} )^2\\ \\ \sqrt{4+2 \sqrt{3}} = \sqrt{(1+ \sqrt{3} )^2} =|1+ \sqrt{3} |=1+ \sqrt{3} [/tex]
