Note that $$(p \operatorname{NOR} q) \equiv (\
eg p \land \
eg q), $$ so that $$ \
eg p \equiv (p \operatorname{NOR} p) $$ and therefore $$ (p \land q) \equiv (\
eg p \operatorname{NOR} \
eg q). $$
Now, $\operatorname{NAND}$ is defined as: $$\begin{align} (p \operatorname{NAND} q) &\equiv \
eg(p \land q), \end{align}$$ so by the above, it's clear how to express it using $\operatorname{NOR}$.