${{\left( 4t+2 \right)}^{3}}+8{{\left( 1-2t \right)}^{3}}$$={{\left( 4t+2 \right)}^{3}}+{{\left( 2-4t \right)}^{3}}$
$=\left( 4t+2+2-4t \right)\left[ {{\left( 4t+2 \right)}^{2}}-\left( 4t+2 \right)\left( 4t-2 \right)+{{\left( 2-4t \right)}^{2}} \right]$
$=4.\left( 16{{t}^{2}}+16t+4-16{{t}^{2}}+4+4-16t+16{{t}^{2}} \right)$
$=4.\left( 16{{t}^{2}}+12 \right)$$=16\left( 4{{t}^{2}}+3 \right)$
${{x}^{3}}+{{y}^{3}}+{{z}^{3}}-3xyz={{\left( x+y \right)}^{3}}-3{{x}^{2}}y-3x{{y}^{2}}+{{z}^{3}}-3xyz$
$=\left[ {{\left( x+y \right)}^{3}}+{{z}^{3}} \right]-3xy\left( x+y+z \right)$
$=\left( x+y+z \right)\left[ {{\left( x+y \right)}^{2}}-z\left( x+y \right)+{{z}^{2}} \right]-3xy\left( x+y+z \right)$
$=\left( x+y+z \right)\left( {{x}^{2}}+2xy+{{y}^{2}}-xz-yz+{{z}^{2}}-3xy \right)$
$=\left( x+y+z \right)\left( {{x}^{2}}+{{y}^{2}}+{{z}^{2}}-xz-yz-xy \right)$