Question : Sum of the factors of $4b^2c^2-(b^2+c^2-a^2)^2$ is:
Option 1: $a+b+c$
Option 2: $2(a+b+c)$
Option 3: $0$
Option 4: $1$
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Correct Answer: $2(a+b+c)$
Solution : $4b^2c^2-(b^2+c^2-a^2)^2$ $=(2bc)^2-(b^2+c^2-a^2)^2$ $=(2bc+b^2+c^2-a^2)(2bc-b^2-c^2+a^2)$ $=((b + c)^2-a^2)(a^2-(b^2+ c^2-2bc))$ $=(b + c + a) (b + c -a) (a^2 -(b-c)^2)$ $=(b+ c+a) (b + c -a) (a + b -c) (a -b + c)$ So, the required sum of factors is given as $b+c+a+b+c-a+a+b-c+a-b+ c=2(a+b+c)$ Hence, the correct answer is $2(a+b+c)$.
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