Simplify the given expression:
$(2 a + b + c)^{2} + (2 a - b - c)^{2}$

8 a^{2} + 2 b^{2} + 2 c^{2} - 4 b c

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8 a^{2} + 2 b^{2} + 2 c^{2} + 4 b c

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8 a^{2} + 2 b^{2} - 2 c^{2} + 4 b c

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8 a^{2} - 2 b^{2} + 2 c^{2} + 4 b c

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Solution

The correct option is B $8 a^{2} + 2 b^{2} + 2 c^{2} + 4 b c$
Consider,
$(2 a + b + c)^{2} + (2 a - b - c)^{2} = [(2 a)^{2} + (b)^{2} + (c)^{2} + 2 \times (2 a) \times (b) + 2 \times (b) \times (c) + 2 \times (c) \times (2 a)]$ + $[(2 a)^{2} + (b)^{2} + (c)^{2} - 2 \times (2 a) \times (b) + 2 \times (b) \times (c) - 2 \times (c) \times (2 a)]$

$= (4 a^{2} + b^{2} + c^{2} + 4 a b + 2 b c + 4 c a)$ + $(4 a^{2} + b^{2} + c^{2} - 4 a b + 2 b c - 4 c a)$
$= 8 a^{2} + 2 b^{2} + 2 c^{2} + 4 b c$

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