TeX source:

\begin{align*} 2^{2(k+1)}-1 &= 2^{2k+2} -1 \\ &= 2^{2k} \cdot 2^2 - 1 \; \; \; \; \; \; \; \;\; \; \; \; \; \; \; \; \mathrm{properties \; of \; exponents}\\ &= 4 \cdot 2^{2k} - 1\\ &= 4 \cdot 2^{2k} -4 +4 -1)\\ &= 4(2^{2k} -1) + 3 \; \; \; \; \; \; \; \; \; \; \; \mathrm{algebra}\\ &= 4(3m) + 3 \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \mathrm{inductive \; hypothesis}\\ &= 3(4m + 1) \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \mathrm{algebra} \end{align*}