Question

The factors of a^2-b^2-4c^2+4d^2-4(ad-bc) are

Answer 1

$Given \:a^{2} - b^{2}-4c^{2} + 4d^{2} - 4(ad-bc) </p><p>$

$= \:a^{2} - b^{2}-4c^{2} + 4d^{2} - 4ad+ 4bc</p><p>$

/* Rearranging the terms, we get */

$= a^{2} + 4d^{2} - 4ad - b^{2} - 4c^{2} + 4bc$

$= (a^{2} + 4d^{2} - 4ad) - (b^{2} +4c^{2} -4bc )$

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

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

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

Therefore.,

$\red{a^{2} - b^{2}-4c^{2} + 4d^{2} - 4(ad-bc) }\\\green { = ( a+b-2c-2d)(a-b+2c-2d) }$

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