Simplify : $(x y + y z)^{2} - (x y - y z)^{2}$

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Solution

$(x y + y z)^{2} - (x y - y z)^{2}$
$\Rightarrow (y (x + z))^{2} - [y (x - z)]^{2}$
$\Rightarrow y^{2} (x + z)^{2} - y^{2} (x - z)^{2}$
$\Rightarrow y^{2} [x^{2} + z^{2} + 2 x z - x^{2} - z^{2} + 2 x z]$
[Since, $(a + b)^{2} = a^{2} + b^{2} + 2 a b, (a - b)^{2} = a^{2} + b^{2} - 2 a b$ ]
$= y^{2} [4 x z]$
$= 4 x y^{2} z$
$∴ (x y + y z)^{2} - (x y - y z)^{2} 4 x y^{2} z$ .
(OR)
$(x y + y z)^{2} - (x y - y z)^{2}$
$= (x y + y z + x y - y z) (x y + y z - x y + y z)$ [Since, $a^{2} - b^{2} = (a + b) (a - b)$ ]
$= (2 x y) (2 y z)$
$= 4 x y^{2} z$

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