Question : Simplify the following expression.
$(a+b+c)^2-(a-b+c)^2+4ac$
Option 1: $4(bc+ac)$
Option 2: $2(ab+bc+ac)$
Option 3: $4(ab+bc+ac)$
Option 4: $4(bc+ab)$
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Correct Answer: $4(ab+bc+ac)$
Solution :
Given:
$(a+b+c)^2-(a-b+c)^2+4 a c$
$= a^2+b^2+c^2+2ab+2bc+2ca-(a^2+b^2+c^2-2ab-2bc+2ca)+4ac$
$= a^2+b^2+c^2+2ab+2bc+2ca-a^2-b^2-c^2+2ab+2bc-2ca+4ac$
$= 4ab+4bc+4ac$
$= 4(ab+bc+ca)$
Hence, the correct answer is $4(ab+bc+ca)$.
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