Solve the equation $y^3 = x^3 + 8x^2 - 6x + 8$ for positive integers $x$ and $y$.
The given solution is:
In the answer, how do we realise that we need to first subtract both sides by $(x+1)^3$ to get the inequality $x+1<y$ and then subtract the entire equation from $(x+3)^3$ to find the second inequality $x+3>y$.
i.e. Why $(x+1)^3$ and $(x+3)^3$ are involved and not $(x\pm n)^3$ where $n$ can be any natural number? How do we arrive at $n = 1$ and $3$ exactly?