Factorise : (a^2- 1) (b^2- 1) + 4ab - Vidyadip

Question

Factorise : $(a^2- 1) (b^2- 1) + 4ab$

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Answer

$(a^2- 1) (b^2- 1) + 4ab$
$= a^2b^2 - a^2 - b^2 + 1 + 4ab$
$= a^2b^2 + 1 + 2ab - a^2 - b^2 + 2ab$
$= ( a^2b^2 + 1 + 2ab ) - ( a^2 + b^2 - 2ab )$
$= ( ab + 1)^2 - ( a - b )^2$
$= [( ab + 1 ) - ( a - b )][( ab + 1 ) + ( a - b )] [ \because a^2 - b^2 = ( a + b )( a - b )]$
$= [ ab + 1 - a + b ][ ab + 1 + a - b ]$

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