Question

Prove that:
$open vertical bar table row straight a cell space space straight a squared end cell cell space space straight b plus straight c end cell row straight b cell space space straight b squared end cell cell space space straight c plus straight a end cell row straight c cell space straight c squared end cell cell space space straight a plus straight b end cell end table close vertical bar space equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis$

Solution

Let $increment space equals space open vertical bar table row straight a cell space space straight a squared end cell cell space space straight b plus straight c end cell row straight b cell space straight b squared end cell cell space space straight c plus straight a end cell row straight c cell space straight c squared end cell cell space straight a plus straight b end cell end table close vertical bar space equals space open vertical bar table row straight a cell space space space straight a squared end cell cell space space straight a plus straight b plus straight c end cell row straight b cell space space straight b squared end cell cell space space straight a plus straight b plus straight c end cell row straight c cell space space straight c squared end cell cell space space straight a plus straight b plus straight c end cell end table close vertical bar comma space by space straight C subscript 3 plus straight C subscript 1$
$equals left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row straight a cell space space straight a squared end cell 1 row straight b cell space straight b squared end cell 1 row straight c cell space straight c squared end cell 1 end table close vertical bar space equals space left parenthesis straight a plus straight b plus straight c right parenthesis space open vertical bar table row 1 cell space straight a end cell cell space straight a squared end cell row 1 cell space straight b end cell cell space straight b squared end cell row 1 cell space straight c end cell cell space straight c squared end cell end table close vertical bar equals space left parenthesis straight a plus straight b plus straight c right parenthesis thin space left parenthesis straight a minus straight b right parenthesis space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space space space space space space space space space space space space space space space space left parenthesis Prove space it right parenthesis$

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