$a + b$ is a factor of which of the following?

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

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a (b - c)^{3} + b (c - a)^{3} + c (a - b)^{3}

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(a + b + c)^{3} - (b + c - a)^{3} - (c + a - b)^{3} - (a + b - c)^{3}

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a (b^{4} - c^{4}) + b (c^{4} - a^{4}) + c (a^{4} + b^{4})

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Solution

The correct option is D $a^{4} (b^{2} - c^{2}) + b^{4} (c^{2} - b^{2}) + c^{4} (a^{2} - b^{2})$
Here we have to follow the options one by one:
in $A$
$\Rightarrow a^{4} (b^{2} - c^{2}) + b^{4} (c^{2} - b^{2}) + c^{4} (a^{2} - b^{2})$
$\Rightarrow a^{4} (b^{2} - c^{2}) - b^{4} (b^{2} - c^{2}) + c^{4} (a^{2} - b^{2})$

$\Rightarrow (a^{4} - b^{4}) (b^{2} - c^{2}) + c^{4} (a^{2} - b^{2})$
$\Rightarrow (a^{4} - b^{4}) (b^{2} - c^{2}) + c^{4} (a^{2} - b^{2})$
$\Rightarrow (a + b) [(a - b) (a^{2} + b^{2}) (b^{2} - c^{2}) + c^{4} (a - b)]$ $(s i n c e, a^{4} - b^{4} = (a^{2} - b^{2}) (a^{2} + b^{2}))$

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