aa + 2 ab + bb . Again ( a - b )( a - b ) gives aa - ab - ab + bb . But two subtractions of ab are equivalent to subtracting 2 ab ; hence the square of a - b is aa -2 ab + bb . Again, ( a + b )( a - b ) gives aa + ab - ab - bb . But the addition and subtraction of ab makes no change; hence the product of a + b and a - b is aa - bb .
Again, the square of a + b + c + d or ( a + b + c + d )( a + b + c + d ) will be found to be aa + 2 ab + 2 ac + 2 ad + bb + 2 bc + 2 bd + cc + 2 cd + dd ; or the rule for squaring such a quantity is: Square the first term, and multiply all that come after by twice that term; do the same