Find the following products

(i) $\left(\frac{x}{2}+2 y\right)\left(\frac{x^{2}}{4}-x y+4 y^{2}\right)=\frac{x}{2}\left(\frac{x^{2}}{4}-x y+4 y^{2}\right)+2 y\left(\frac{x^{2}}{4}-x y+4 y^{2}\right)$

$=\frac{x}{2}\left(\frac{x^{2}}{4}\right)+\frac{x}{2}(-x y)+\frac{x}{2}\left(4 y^{2}\right)+2 y\left(\frac{x^{2}}{4}\right)+2 y(-x y)+2 y\left(4 y^{2}\right)$

$=\frac{x^{3}}{8}-\frac{x^{2} y}{2}+2 x y^{2}+\frac{x^{2} y}{2}-2 x y^{2}+8 y^{3}$

$=\frac{x^{3}}{8}+8 y^{3}$

(ii) $\left(x^{2}-1\right)\left(x^{4}+x^{2}+1\right)=x^{2}\left(x^{4}+x^{2}+1\right)-1\left(x^{4}+x^{2}+1\right)$

$=\left[x^{2}\left(x^{4}\right)+x^{2}\left(x^{2}\right)+x^{2}(1)-1(x)^{4}-1\left(x^{2}\right)-1(1)\right]$

$=x^{6}+x^{4}+x^{2}-x^{4}-x^{2}-1=x^{6}-1$