Powers of Polynomials

Squaring the two given equations, we get:

$m^6-24m^4n^2+144m^2n^4=1600 \hspace{5cm}(1)$

$16n^6-24m^2n^4+9m^4n^2=100 \hspace{5cm}(2)$

Multiplying equation $(2)$ by $4$ , we get:

$\displaystyle 64n^6-96m^2n^4+36m^4n^2=400 \hspace{5cm}(3)$

Adding equations $(1)$ and $(3)$ , we get $m^6+12m^4n^2+48m^2n^4+64n^6=(m^2+4n^2)^3=2000$ . Therefore, $m^2+4n^2=\sqrt[3]{2000}=\boxed{10\sqrt[3]{2}}.$