Question:
Find the value
$a^{2}-b^{2}+2 b c-c^{2}$
Solution:
$a^{2}-\left(b^{2}-2 b c+c^{2}\right)$
Using the identity $(a-b)^{2}=a^{2}+b^{2}-2 a b$
$=a^{2}-(b-c)^{2}$
Using the identity $a^{2}-b^{2}=(a+b)(a-b)$
$=(a+b-c)(a-(b-c))$
$=(a+b-c)(a-b+c)$
$\therefore a^{2}-b^{2}+2 b c-c^{2}=(a+b-c)(a-b+c)$
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