Root Algebra

$\sqrt{ {\color{#3D99F6}a + b +c} + \sqrt{\color{#E81990}2ac +2bc} }\times \sqrt{ {\color{#3D99F6}a+b+c} - \sqrt{{\color{#E81990}2ac+2bc}}} = 24$

$\implies \sqrt{ ({\color{#3D99F6}a + b + c})^2 - (\sqrt{ {\color{#E81990}2ac + 2bc} })^2} = 24 \quad \quad (\text{Since it is in the form :} (a + b)(a - b) = a^2 - b^2)$

$\implies \sqrt{a^2 + b^2 + c^2 + 2ab + \cancel{2bc} + \cancel{2ac} - \cancel{2ac} - \cancel{2bc}} = 24$

$\implies a^2 + b^2 + c^2 + 2(ab) = 576$

$\implies a^2 + b^2 + c^2 + 2(6) = 576$

$\implies a^2 + b^2 + c^2 = 576 - 12 = 564$