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$x^{2} + 4 y^{2} + 25 z^{2} - 4 x y - 20 y z + 10 z x =$

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{(x - 2 y - 5 z)}^{2}

{(x - 2 y + 5 z)}^{2}

{(x + 2 y + 5 z)}^{2}

{(x + 2 y - 5 z)}^{2}

answer is C.

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Detailed Solution

x^{2} + 4 y^{2} + 25 z^{2} - 4 x y - 20 y z + 10 z x

$= x^{2} + {(- 2 y)}^{2} + {(5 z)}^{2} + 2 (x) (- 2 y) + 2 (- 2 y) (5 z) + 2 (5 z) (x)$

This is in the form of

$a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a = {(a + b + c)}^{2}$

$= {(x - 2 y + 5 z)}^{2}$

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