I hope it was purely trial and error and there was no systematic way to arrive at this expression as the textbook didn't provide the work and I'm not able to see how one cooks this up systematically. For example, how would one go about cooking such an expression for showing that the set of rational numbers satisfying $p^2<3$ has no largest number ? I've tried this by replacing $2$ in the given expression by $3$. It doesn't work. Appreciate any ideas. Thanks !
A nice proof.
Replace $2$ by $3$.
$(3)$ becomes $\displaystyle q=p-\frac{p^2-3}{p+3}=\frac{3p+3}{p+3}$
$(4)$ becomes $\displaystyle q^2-3=\frac{6(q^2-3)}{(q+3)^2}$