\[(p + 1)^{2} - (p + 2)^{2} =\]
\[= \left( p + 1 - (p + 2) \right)(p + 1 + p + 2) =\]
\[= (p + 1 - p - 2)(2p + 3) =\]
\[= - 1 \cdot (2p + 3) = - 2p - 3.\]