Factorize $a^2 - b^2 + c^2 - d^2 - 2 (ac+bd)$.

Given: The expression: $a^2-b^2+c^2-d^2-2( ac-bd)$.

To do: To factorize $a^2-b^2+c^2-d^2-2( ac-bd)$.

$a^2-b^2+c^2-d^2-2( ac-bd)$

$=a^2-b^2+c^2-d^2-2ac-2bd$

$=( a^2+c^2-2ac)-( b^2+d^2-2bd)$

$=( a-c)^2-( b-d)^2$

$=( a-c+b-d)\times( a-c-b +d)$         [$\because a^2-b^2=(a+b)(a-b)$]

$=( a+b-c-d)\times( a-b-c+d)$