The correct option is C (a+b+c)2(a−b)(b−c)(a−c)
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∣∣ab+ca3bc+ab3ca+bc3∣∣
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C2→C2+C1
=∣∣
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∣∣aa+b+ca3ba+b+cb3ca+b+cc3∣∣
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=(a+b+c)∣∣
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∣∣a1a3b1b3c1c3∣∣
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R1→R1−R2,R2→R2−R3
=(a+b+c)∣∣
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∣∣a−b0a3−b3b−c0b3−c3c1c3∣∣
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=(a+b+c)(a−b)(b−c)∣∣
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∣∣10a2+b2+ab10b2+c2+bcc1c3∣∣
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=(a+b+c)(a−b)(b−c)[c2−a2+b(c−a)]
=(a+b+c)2(a−b)(b−c)(c−a)