NMTC

$a^3+4a-8=0$ $\Rightarrow\space a^3=-4(a-2)$ $\Rightarrow\space a^6=(a^3)^2=16(a^2+4-4a)=16a^2+64-64a$ $\Rightarrow\space a^7=a(a^6)=a(16a^2+64-64a)=16a^3+64a-64a^2$ $\therefore\space a^7+64a^2=16a^3+64a=16(a^3+4a)=16\times8=128.$

$\begin{aligned} a^3 + 4a-8 &=0 \\ a^7 + 4a^5-8a^4 &=0 \\ a^7 + 4a^3(a^2-2a) &=0 \\ a^7 + 4(-4a+8)(a^2-2a) &=0 \\ a^7 - 16a(a-2)^2 &=0 \\ a^7 - 16a(a^2-4a+4) &=0 \\ a^7 - 16a^3 + 64a^2-64a &=0 \\ a^7 + 64a^2 &= 16(a^3 +4a) \\ &= 16(-4a+8+4a) \\ &= \boxed{128} \end{aligned}$