so irrespective of the fact the book did not factor completely would $\displaystyle (x^3-\frac{1}{{\color{red}8}})(x^3{\color{red}+}\frac{1 }{{\color{red}8}})$ be correct? If so how do you arrive at that particular answer?

To continue, you would have to know the model for the difference and sum of cubes. Is that part of your lesson?
Also could someone walk me through this problem as well $\displaystyle 16x^4-1=(2x)^4-1^4=(4x^2-1)(4x^2+1)=(2x-1)(2x+1)(4x^2+1)$

Lost in this one as well unfortunately

. I understand breaking down $\displaystyle 16x^4-1=(2x)^4-1^4$ but how that yields $\displaystyle (4x^2-1)(4x^2+1)$ I am not certain.