$Prove space that open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space equals straight a cubed plus straight b cubed plus straight c cubed space minus 3 abc$

Let space space space increment space space equals space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space space space space space space space space space space space space space space equals space straight a space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space minus straight b space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space plus straight c space space open vertical bar table row straight a straight b straight c row cell straight a minus straight b end cell cell space space space straight b minus straight c end cell cell space space space straight c minus straight a end cell row cell straight b plus straight c end cell cell space space straight c plus straight a end cell cell space space space straight a plus straight b end cell end table close vertical bar space space space space space space space space space space space space space space equals space straight a space open square brackets table row cell left parenthesis straight b minus straight c right parenthesis end cell cell left parenthesis straight a plus straight b right parenthesis minus end cell cell left parenthesis straight c plus straight a right parenthesis end cell cell left parenthesis straight c minus straight a right parenthesis end cell end table close square brackets space minus straight b space open square brackets table row cell left parenthesis straight a minus straight b right parenthesis end cell cell left parenthesis straight a plus straight b right parenthesis minus end cell cell left parenthesis straight b plus straight c right parenthesis end cell cell left parenthesis straight c minus straight a right parenthesis end cell end table close square brackets space plus straight c space space open square brackets table row cell left parenthesis straight a minus straight b right parenthesis end cell cell left parenthesis straight c plus straight a right parenthesis minus end cell cell left parenthesis straight b minus straight c right parenthesis end cell cell left parenthesis straight b plus straight c right parenthesis end cell end table close square brackets space space space space space space space space space space space space space space equals space straight a space open square brackets straight a table row cell straight b plus straight b squared end cell cell negative ca minus bc end cell cell straight c squared plus ca minus ca plus end cell cell straight a squared end cell end table close square brackets space minus straight b space open square brackets straight a squared space minus straight b to the power of 2 space end exponent minus bc plus ab minus straight c squared plus ca close square brackets plus straight c open square brackets table row cell ac plus straight a squared end cell end table minus bc minus ab minus straight b squared space plus straight c squared close square brackets space space space space space space space space space space space space space space equals space straight a space open square brackets straight a squared table row cell plus straight b squared end cell cell negative straight c squared minus ab minus bc minus ca end cell plus cell straight a squared end cell end table close square brackets space minus straight b space open square brackets straight a squared space minus straight b to the power of 2 space end exponent minus straight c squared plus ab minus bc plus ca close square brackets plus straight c open square brackets table row cell straight a squared minus end cell end table straight b squared plus straight c squared minus ab minus bc plus ca close square brackets space space space space space space space space space space space space space space equals space space straight a cubed space plus ab squared space minus ac squared space plus straight a squared space straight b minus abc minus straight a squared space straight c minus straight a squared space straight b plus straight b cubed space plus straight b squared space straight c minus ab squared space plus bc squared space plus bc cubed space minus abc plus straight a squared space straight c space minus straight b squared straight c space plus straight c cubed space minus abc minus bc squared space plus straight c to the power of 2 space end exponent straight a space space space space space space space space space space space space space space equals space space straight a cubed space plus straight b cubed space plus straight c cubed space minus 3 abc

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