Question

$show space that open vertical bar table row straight a cell space space space space space space space space space straight a plus straight b end cell cell straight a plus straight b plus straight c end cell row cell 2 straight a end cell cell space space space space space space space space 3 straight a plus 2 straight b end cell cell space space 4 straight a plus 3 straight b plus 2 straight c end cell row cell 3 straight a end cell cell space space space space space 6 straight a plus straight b end cell cell space space space space space 10 straight a plus 6 straight b plus 3 straight c end cell end table close vertical bar space equals straight a cubed$

Solution

Let space space increment space equals space open vertical bar table row straight a cell space space space space space space space space space straight a plus straight b end cell cell straight a plus straight b plus straight c end cell row cell 2 straight a end cell cell space space space space space space space space 3 straight a plus 2 straight b end cell cell space space 4 straight a plus 3 straight b plus 2 straight c end cell row cell 3 straight a end cell cell space space space space space 6 straight a plus straight b end cell cell space space space space space 10 straight a plus 6 straight b plus 3 straight c end cell end table close vertical bar space space space space space space space space space space space equals space open vertical bar table row straight a cell space space space space space space space straight a plus straight b space space space space space end cell cell straight a plus straight b plus straight c end cell row 0 straight a cell 2 straight a plus straight b end cell row 0 cell 3 straight a end cell cell 7 straight a plus 3 straight b end cell end table close vertical bar space comma space by space straight R subscript 2 space end subscript minus 2 straight R subscript 1 space comma space straight R subscript 3 space minus 3 space straight R subscript 1 space space space space space space space space space space equals space straight a space open vertical bar table row straight a cell space space 2 straight a plus straight b end cell row cell 3 straight a end cell cell space space space 7 straight a plus 3 straight b end cell end table close vertical bar space comma by space expanding space with space first space column space space space space space space space space space space equals space straight a space space open square brackets table row cell 7 straight a squared end cell end table plus 3 ab minus 6 straight a squared space minus 3 ab close square brackets space equals straight a left parenthesis straight a squared right parenthesis space equals straight a cubed

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