Factorise: ${(x + 2 y)}^{2} - 4 {(2 x - y)}^{2}$

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Solution

${(x + 2 y)}^{2} - 4 {(2 x - y)}^{2}$
$= {(x + 2 y)}^{2} - {(2 (2 x - y))}^{2}$
$= {(x + 2 y)}^{2} - {(4 x - 2 y)}^{2}$ is of the form $a^{2} - b^{2} = (a - b) (a + b)$
$= (x + 2 y - 4 x + 2 y) (x + 2 y + 4 x - 2 y)$
$= (- 3 x + 4 y) (5 x)$
$= 5 x (4 y - 3 x)$

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