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ANSWER
[tex]{(m + 2)}( {n}^{2} + 2n + m + 1)[/tex]

EXPLANATION

The given expression is

[tex] {(m + 2)}^{2} - (m + 2) + mn(n + 2) + 2n(n + 2)[/tex]

We factor to get,

[tex] = {(m + 2)}(m + 2 - 1) +n (n + 2) (m+ 2)[/tex]

This implies that,

[tex] = {(m + 2)}(m + 1) +n (n + 2) (m+ 2)[/tex]

We factor further to obtain,

[tex] = {(m + 2)}((m + 1) +n (n + 2))[/tex]

or

[tex] = {(m + 2)}( {n}^{2} + 2n + m + 1)[/tex]

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