Value

$\begin{aligned} x & = 16\left(m^2+n^2\right)^2 + 56\left(m^2+n^2\right)\left(3m^2-2n^2\right) + 49\left(3m^2-2n^2\right)^2 \\ & = 4^2\left(m^2+n^2\right)^2 + 2\cdot 4 \left(m^2+n^2\right)\cdot 7 \left(3m^2-2n^2\right) + 7^2\left(3m^2-2n^2\right)^2 \\ & = \left(4\left(m^2+n^2\right) + 7 \left(3m^2-2n^2\right)\right)^2 \\ & = \left(4m^2+4n^2 + 21 m^2- 14n^2 \right)^2 \\ & = \left(25 m^2- 10n^2 \right)^2 \\ & = \left(900- 490 \right)^2 \\ & = 410^2 \\ & = (400+10)^2 \\ & = 160000 + 8000 + 100 \\ & = \boxed{168100} \end{aligned}$