`\sqrt{12 + 3 \sqrt{3} + \sqrt{4 + 2 \sqrt{3}} - 2 \sqrt{3}`

` = \sqrt{12 + 3 \sqrt{3} + \sqrt{3 + 2 \sqrt{3} + 1}} - 2 \sqrt{3}`

` = \sqrt{12 + 3 \sqrt{3} + \sqrt{ (\sqrt{3})^2 + 2 . \sqrt{3} . 1 + 1^2}} - 2 \sqrt{3}`

` = \sqrt{12 + 3 \sqrt{3} + \sqrt{ (\sqrt{3} + 1)^2 } - 2 \sqrt{3}`

` = \sqrt{12 + 3 \sqrt{3}  + |\sqrt{3} + 1|} - 2 \sqrt{3}`

` = \sqrt{12 + 3 \sqrt{3} + \sqrt{3} + 1} - 2 \sqrt{3}`

` = \sqrt{12 + 4 \sqrt{3} + 1} - 2 \sqrt{3}`

`=  \sqrt{12 + 2 . 2\sqrt{3} . 1 + 1^2} - 2\sqrt{3}`

` = \sqrt{ (2\sqrt{3} + 1)^2} - 2\sqrt{3}`

` = |2 \sqrt{3} +1 | - 2 \sqrt{3}`

` = 2\sqrt{3} + 1 - 2\sqrt{3}`

` = 1`