Simplify: ${(3 m + 5 n)}^{2} - {(2 n)}^{2}$

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Solution

We have: ${(3 m + 5 n)}^{2} - {(2 n)}^{2}$

Using the identity: $(x + y)^{2} = x^{2} + y^{2} + 2 x y$ for the above expression we get,

${(3 m + 5 n)}^{2} - {(2 n)}^{2} = [{(3 m)}^{2} + {(5 n)}^{2} + (2 \times 3 m \times 5 n)] - 4 n^{2}$

$= 9 m^{2} + 25 n^{2} + 30 m n - 4 n^{2} = 9 m^{2} + 21 n^{2} + 30 m n$

$= 3 (3 m^{2} + 7 n^{2} + 10 m n)$

Hence, ${(3 m + 5 n)}^{2} - {(2 n)}^{2} = 3 (3 m^{2} + 7 n^{2} + 10 m n)$ .

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