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

${4 x y}^{2}$

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$4 x y^{2} z$

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$4 x z$

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$- 4 x y^{2} z$

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Solution

The correct option is B
$4 x y^{2} z$

Using the identity,
${(a + b)}^{2} = a^{2} + b^{2} + 2 a b,$

we get,
${(x y + y z)}^{2} = x^{2} y^{2} + y^{2} z^{2} + 2 x z y^{2} . . . (1)$

Using the identity,
${(a - b)}^{2} = a^{2} + b^{2} - 2 a b$

we get,
${(x y - y z)}^{2} = x^{2} y^{2} + y^{2} z^{2} - 2 x z y^{2} . . . (2)$

Subtracting (2) from (1), we get

$(x^{2} y^{2} + y^{2} z^{2} + 2 x z y^{2}) - (x^{2} y^{2} + y^{2} z^{2} - 2 x z y^{2})$
$= 2 x z y^{2} + 2 x z y^{2}$
$= 4 x y^{2} z$

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