Factorise: $a^{2} + 2 a b + b^{2} - 9 c^{2}$

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Solution

Applying the identities: $a^{2} - b^{2} = (a + b) (a - b)$ and $(a + b)^{2} = a^{2} + b^{2} + 2 a b$

Using the above identities, the equation $a^{2} + 2 a b + b^{2} - 9 c^{2}$ can be factorised as follows:

$a^{2} + 2 a b + b^{2} - 9 c^{2} = (a + b)^{2} - 9 c^{2} = (a + b)^{2} - (3 c)^{2} = (a + b + 3 c) (a + b - 3 c)$

Hence, $a^{2} + 2 a b + b^{2} - 9 c^{2} = (a + b + 3 c) (a + b - 3 c)$

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