Factorise: $l^{4} + 4 l^{2} m^{2} + 4 m^{4}$

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Solution

We need to factorise $l^{4} + 4 l^{2} m^{2} + 4 m^{4}$
It can be written as
$l^{4} + 4 l^{2} m^{2} + 4 m^{4} = l^{4} + 2 l^{2} m^{2} + 2 l^{2} m^{2} + 4 m^{4}$
$= l^{2} (l^{2} + 2 m^{2}) + 2 m^{2} (l^{2} + 2 m^{2})$
$= (l^{2} + 2 m^{2}) (l^{2} + 2 m^{2})$
$= (l^{2} + 2 m^{2})^{2}$
Thus factors are $(l^{2} + 2 m^{2}), (l^{2} + 2 m^{2})$ .

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