$8^2 + 4^3 = 64 + 4\times4\times4$. We can use $8$s again to work out $4\times4\times4$:
$4\times4\times4= 4\times (2\times2)\times 4 = \left(4\times2\right)\times\left(2\times4\right) = 8\times8=64.$
So $8^2+4^3 = 64+64=128.$
And $8\times4\times2 = 8\times8=64.$
Clearly $64<126<128<130$, so $8\times4\times2<2\left(4^3\right)-2<8^2+4^3<2^7+2$