Simplify: $\;$ $\left[\dfrac{1}{\left(m + n\right)^2} \left(\dfrac{1}{m^2} + \dfrac{1}{n^2}\right) + \dfrac{2}{\left(m + n\right)^3} \left(\dfrac{1}{m} + \dfrac{1}{n}\right)\right] m^2 n^2$
$\left[\dfrac{1}{\left(m + n\right)^2} \left(\dfrac{1}{m^2} + \dfrac{1}{n^2}\right) + \dfrac{2}{\left(m + n\right)^3} \left(\dfrac{1}{m} + \dfrac{1}{n}\right)\right] m^2 n^2$
$= \left[\dfrac{1}{\left(m + n\right)^2} \left(\dfrac{m^2 + n^2}{m^2 n^2}\right) + \dfrac{2}{\left(m + n\right)^3} \left(\dfrac{m + n}{mn}\right)\right] m^2 n^2$
$= \left[\dfrac{1}{\left(m + n\right)^2} \left(\dfrac{m^2 + n^2}{m^2 n^2}\right) + \dfrac{2}{\left(m + n\right)^2} \times \dfrac{1}{mn}\right] m^2 n^2$
$= \dfrac{m^2 + n^2}{\left(m + n\right)^2} + \dfrac{2 mn}{\left(m + n\right)^2}$
$= \dfrac{m^2 + n^2 + 2mn}{\left(m + n\right)^2}$
$= \dfrac{\left(m + n\right)^2}{\left(m + n\right)^2}$
$= 1$