Factorise the expression $x^{2} + 2 x y + y^{2} - 4 z^{2}$

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Solution

The first three terms of the expression is in the form $(x + y)^{2}$ and the fourth term is a perfect square.
Hence, $x^{2} + 2 x y + y^{2} - 4 z^{2} = (x + y)^{2} - (2 z)^{2}$
$= [(x + y) + 2 z] [(x + y) - 2 z]$
$= (x + y + 2 z) (x + y - 2 z)$

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