a+b+2x+2ab+(a+b)x+x2=

a+b+2x+2ab+(a+b)x+x2=(a+x)+((b+x)+2(a+x)(b+x)=(a+x)2+b+x2+2.a+xb+x=(a+x+b+x)2=a+x+b+x

a+b+2x+2ab+(a+b)x+x2=

a+b+2x+2ab+(a+b)x+x2=(a+x)+((b+x)+2(a+x)(b+x)=(a+x)2+b+x2+2.a+xb+x=(a+x+b+x)2=a+x+b+x

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$\sqrt{a + b + 2 x + 2 \sqrt{ab + (a + b) x + x^{2}}} =$

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$\sqrt{a + b + 2 x}$

$\sqrt{a} + \sqrt{b} + \sqrt{x}$

$\sqrt{a + x} + \sqrt{b + x}$

$\sqrt{a + b + x} + \sqrt{x}$

answer is C.

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Detailed Solution

$\begin{array}{l} \sqrt{a + b + 2 x + 2 \sqrt{a b + (a + b) x + x^{2}}} \\ = \sqrt{(a + x) + ((b + x) + 2 \sqrt{(a + x) (b + x)}} \\ = \sqrt{{(\sqrt{a + x})^{2} + \sqrt{b + x})}^{2} + 2 . \sqrt{a + x} \sqrt{b + x}} \\ = \sqrt{{(\sqrt{a + x} + \sqrt{b + x})}^{2}} \end{array}$
$= \sqrt{a + x} + \sqrt{b + x}$