Simplify: ${(x + 2 y + 3 z + r)}^{2} + {(x + 2 y + 3 z - r)}^{2}$

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Solution

Consider, ${(x + 2 y + 3 z + r)}^{2} + {(x + 2 y + 3 z - r)}^{2}$
$= ((x + 2 y + 3 z) + r)^{2} + ((x + 2 y + 3 z) - r)^{2}$
$= (x + 2 y + 3 z)^{2} + r^{2} + 2 r (x + 2 y + 3 z) + (x + 2 y + 3 z)^{2} + r^{2} - 2 r (x + 2 y + 3 z)$
$= 2 r^{2} + 2 (x + 2 y + 3 z)^{2}$
$= 2 (r^{2} + (x + 2 y + 3 z)^{2})$
$= 2 (r^{2} + x^{2} + (2 y)^{2} + (3 z)^{2} + 2 (2 x y) + 2 (3 x z) + 2 (6 y z))$
$= 2 (x^{2} + 4 y^{2} + 9 z^{2} + 4 x y + 12 y z + 6 x z + r^{2})$

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