Solution:
For any integer $n \geq 0$, the given implies $x^{n+3} = -4x^{n+1} + 8x^{n}$, so we can rewrite any such power of $x$ in terms of lower powers. Carrying out this process iteratively gives
$$\begin{aligned} x^{7} & = -4x^{5} + 8x^{4} \\ & = 8x^{4} + 16x^{3} - 32x^{2} \\ & = 16x^{3} - 64x^{2} + 64x \\ & = -64x^{2} + 128. \end{aligned}$$
Thus, our answer is $128$.