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

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Solution

Simplify the algebric expression:
Given: Simplify: ${(m^{2} - n^{2} m)}^{2} + 2 m^{3} n^{2}$
Using the identities
${(a + b)}^{2} = a^{2} + b^{2} + 2 ab$
${(a - b)}^{2} = a^{2} + b^{2} - 2 ab$
${(m^{2} - n^{2} m)}^{2} + 2 m^{3} n^{2}$
Consider ${(m^{2} - n^{2} m)}^{2}$
Using identity
${(a - b)}^{2} = a^{2} + b^{2} - 2 ab$
${(m^{2} - n^{2} m)}^{2}$ $=$ ${({(m)}^{2})}^{2} + {(n^{2} m)}^{2} - 2 \times m^{2} \times n^{2} m$
$\Rightarrow$ ${(m^{2} - n^{2} m)}^{2}$ $=$ $m^{4} + n^{4} m^{2} - 2 m^{3} n^{2}$
Now consider
${(m^{2} - n^{2} m)}^{2} + 2 m^{3} n^{2}$ $=$ $m^{4} + n^{4} m^{2} - 2 m^{3} n^{2} + 2 m^{3} n^{2}$
$\Rightarrow$ ${(m^{2} - n^{2} m)}^{2} + 2 m^{3} n^{2}$ $=$ $m^{4} + n^{4} m^{2}$
Hence, ${(m^{2} - n^{2} m)}^{2} + 2 m^{3} n^{2}$ is equal to $m^{4} + n^{4} m^{2}$

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