Algebraic steps to translate this expression into the pictured form? ![enter image description here]( What are the intermediate steps to arrive at the second form?

Just considering the numerators,

$$\begin{eqnarray*} n(n+1)(2n+1)+6(n+1)^2 &=& (n+1)n(2n+1)+6(n+1)(n+1)\\\ &=& (n+1)\bigl[n(2n+1) + 6(n+1)\bigr]\\\ &=& (n+1)(2n^2+ n + 6n+6) \end{eqnarray*}$$