$\begin{array}{l}\sqrt{a + b +c + 2\sqrt{ac + bc}}+ \sqrt{a + b +c - 2\sqrt{ac + bc}}\\ = \sqrt{(a + b) +c + 2\sqrt{a + b}.\sqrt c}+ \sqrt{(a + b) +c - 2\sqrt{a + b}.\sqrt c}\\ = |\sqrt{a + b} + \sqrt c| + |\sqrt{a + b} - \sqrt c|\\ = \sqrt{a + b} + \sqrt c + |\sqrt{a + b} - \sqrt c|\\ =\left[\begin{array}{l}\sqrt{a + b} + \sqrt c + \sqrt{a + b} - \sqrt c \\\sqrt{a + b} + \sqrt c - (\sqrt{a + b} - \sqrt c)\end{array}\right.\\ = \left[\begin{array}{l}2\sqrt{a + b}\qquad ( a + b \geq c \geq 0)\\2\sqrt c\qquad \qquad(c > a + b \geq 0)\end{array}\right.\\\end{array}$
