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Answer 1

EXPLANATION.

$\sf \: \implies \: \sqrt{2x + \sqrt{2x + 4} } = 4$

$\sf \: \implies \: 2x \: + \sqrt{2x + 4} = (4) {}^{2} \\ \\ \sf \: \implies \: 2x \: + \sqrt{2x + 4} = 16 \\ \\ \sf \: \implies \: \sqrt{2x + 4} = 16 - 2x \\ \\ \sf \: \implies \: 2x \: + \: 4 \: = (16 - 2x) {}^{2}$

$\sf \: \implies \: 2x \: + \: 4 \: = \: ( {16})^{2} + {2x}^{2} - 2(16)(2x) \\ \\ \sf \: \implies \: 2x \: + \: 4 \: = 256 + 4 {x}^{2} - 64x \\ \\ \sf \: \implies \: 4 {x}^{2} - 64x + 256 - 2x - 4 = 0 \\ \\ \sf \: \implies \: 4 {x}^{2} - 66x + 252 = 0 \\ \\ \sf \: \implies \:2(2 {x}^{2} - 33x + 126) = 0$

$\sf \: \implies \: {2x}^{2} - 33x + 126 = 0 \\ \\ \sf \: \implies \: 2 {x}^{2} - 21x - 12x + 126 = 0 \\ \\ \sf \: \implies \: x(2x - 21) - 6(2x - 21) = 0 \\ \\ \sf \: \implies \: (x - 6)(2x - 21) = 0 \\ \\ \sf \: \implies \: x \: = 6 \: \: and \: \: x \: = \frac{21}{2}$