Show language is not regular
For these hints, I'm assuming that you know (and are allowed to use) the fact that $\{a^nb^n\mid n\ge 0\}$ is not regular.

For (1), if $L_1$ was regular, then since regular languages are closed under intersection, we'd have $L_1\cap b^*c^*$ was regular. What's that language?

For (2), if $L_2$ was regular, then since regular languages are closed under complementation, we'd have $\overline{L_2}$ was regular, and so $\overline{L_2}\cap a^*b^*$ was regular. What's that language?