Find the Missing Sign in m³+2m²n=4m²+n²: Comparing Perfect Square Expressions

To solve this problem, we aim to relate the expression $m(m+n)^2$ and $(2m+n)^2$ given the equation $m^3 + 2m^2n = 4m^2 + n^2$.

The key step is to understand that due to the given constraint equation, without loss of generality or additional specific values for $m$ and $n$, a direct comparison to determine the sign $?$ is not feasible under the constraint $m^3 + 2m^2n = 4m^2 + n^2$. The nature of the relationship defined by the constraint tells us that the relationship between these expressions might hold under special conditions, but projecting one side as definitively greater, less, or equal is not straightforward.

Given this complexity and the lack of further simplifiable information directly provided by the equation, it is not possible to calculate or visually confirm which sign should be appropriately used without more specific values or further simplification details.

Therefore, the answer is: It is not possible to calculate.