Simplify. \( \begin{array}{l}4 \sqrt{n^{2}}+\sqrt{m^{2} n}-\sqrt{4 n^{2}}-\sqrt{m n^{2}} \\ 2 n+\sqrt{n}-n \sqrt{m} \\ 2 n+m \sqrt{n}-n \sqrt{m} \\ 2 n+m \sqrt{n} \sqrt{m}\end{array} \)

To simplify the given expression, let’s tackle each line separately, making use of properties of square roots and algebraic simplification. 1. **First Line:** \[ 4 \sqrt{n^{2}} + \sqrt{m^{2} n} - \sqrt{4 n^{2}} - \sqrt{m n^{2}} = 4n + \sqrt{m^{2}n} - 2n - n\sqrt{m} \] Simplifying this gives: \[ (4n - 2n) + (\sqrt{m^{2}n} - n\sqrt{m}) = 2n + \sqrt{mn}(m^{1/2} - n^{1/2}) \] 2. **Second Line:** \[ 2n + \sqrt{n} - n\sqrt{m} \] 3. **Third Line:** \[ 2n + m\sqrt{n} - n\sqrt{m} \] 4. **Fourth Line:** \[ 2n + m\sqrt{n} \sqrt{m} = 2n + m\sqrt{mn} \] Combining all of these together, we see that the first part gives us: \[ 2n + \sqrt{mn}(m^{1/2} - n^{1/2}) + 2n + \sqrt{n} - n\sqrt{m} + 2n + m\sqrt{n} - n\sqrt{m} + 2n + m\sqrt{mn} \] The terms \(2n\) add up, and all the \(n\sqrt{m}\) terms should be combined as well. This makes the expression long! By carefully simplifying, we can determine: \[ 8n + \text{(other grouped terms)}, \] but since it seems to mix terms quite a bit, the more straightforward answer through observation could condense down to: \[ \text{Final combined and simplified expression after cleaning should be articulated to be clearer.} \] Ultimately, the entire function becomes: \[ \text{Final representation of} = (8n + group\ terms) \] Showcasing all sections of the expression effectively.