Verify whether the given value of $x = \frac{-2}{m+n}$ is a solution of the quadratic equation $(m+n)^{2} x^{2} + (m+n) x - 2 = 0$ or not.

To verify whether $x = \frac{-2}{m+n}$ is a solution, we substitute it into the quadratic equation.
Given equation:
$(m+n)^2 x^2 + (m+n)x - 2 = 0$
Substitute $x = \frac{-2}{m+n}$:
$(m+n)^2 \left(\frac{-2}{m+n}\right)^2 + (m+n)\left(\frac{-2}{m+n}\right) - 2 = 0$
Simplifying:
$(m+n)^2 \cdot \frac{4}{(m+n)^2} + (m+n)\cdot \frac{-2}{m+n} - 2 = 0$
$4 - 2 - 2 = 0$
$0 = 0$
Since both sides are equal, $x = \frac{-2}{m+n}$ is a solution of the given quadratic equation.