Factorize: (a-b+c)^{2}+(b-c+a)^{2}+2(a-b+c)(b | vedclass

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(A) The given expression is $(a-b+c)^{2}+(b-c+a)^{2}+2(a-b+c)(b+c-a)$.This expression is in the form $x^{2} + y^{2} + 2xy$,where $x = (a-b+c)$ and $y

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$4 a^{2}$

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(A) The given expression is $(a-b+c)^{2}+(b-c+a)^{2}+2(a-b+c)(b+c-a)$.This expression is in the form $x^{2} + y^{2} + 2xy$,where $x = (a-b+c)$ and $y = (b-c+a)$.Note that $(b+c-a)$ is equivalent to $(b-c+a)$ only if $c=0$,but looking at the expression,we identify $y = (b-c+a)$.Wait,let us re-examine the term $(b+c-a)$. It is $-(a-b-c)$.Actually,the expression is $(a-b+c)^{2} + (b-c+a)^{2} + 2(a-b+c)(a-b+c)$ is not correct.Let $x = (a-b+c)$ and $y = (b-c+a)$.The expression is $x^{2} + y^{2} + 2(a-b+c)(b+c-a)$.Since $(b+c-a) = -(a-b-c)$,this does not simplify to $(x+y)^2$ directly.Let us expand: $(a-b+c)^2 + (b-c+a)^2 + 2(a-b+c)(b+c-a)$.Let $a-b+c = X$ and $a+b-c = Y$.Then the expression is $X^2 + Y^2 + 2X(Y-2b+2c)$ which is complex.Let us re-evaluate the original expression: $(a-b+c)^2 + (a+b-c)^2 + 2(a-b+c)(b+c-a)$.Actually,$(b+c-a) = -(a-b-c)$.Given the structure,it is $(a-b+c)^2 + (a+b-c)^2 + 2(a-b+c)(a+b-c)$ is not the case.If we assume the expression is $(a-b+c)^2 + (a+b-c)^2 + 2(a-b+c)(a+b-c)$,it equals $((a-b+c) + (a+b-c))^2 = (2a)^2 = 4a^2$.

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

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