Simplify: $(a b + b c)^{2} - 2 a b^{2} c$ .

a b + b c

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a^{2} b^{2} - b^{2} c^{2}

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

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

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Solution

Given, $(a b + b c)^{2} - 2 a b^{2} c$ .
We know,
$(a + b)^{2} = a^{2} + 2 a b + b^{2}$ .
Then, $(a b + b c)^{2} - 2 a b^{2} c$
$= (a b)^{2} + 2 (a b) (b c) + (b c)^{2} - 2 a b^{2} c$
$= a^{2} b^{2} + 2 a b^{2} c + b^{2} c^{2} - 2 a b^{2} c$
$= a^{2} b^{2} + 2 a b^{2} c - 2 a b^{2} c + b^{2} c^{2}$ [Since, $2 a b^{2} c, - 2 a b^{2} c$ are like terms]
$= a^{2} b^{2} + b^{2} c^{2}$ . [Since, $2 a b^{2} c - 2 a b^{2} c = 0$ ]
Therefore, option $C$ is correct.

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