#### Answer

$z_1^2 - 1$ = $(a^2 - b^2 - 1) + 2abi$
$z_2^2 - 1$ = $(a^2 - b^2 - 1) - 2abi$

#### Work Step by Step

For $z_1 = a + bi$,
$z_1^2 - 1$
= $(a + bi)^2 - 1$
= $a^2 + 2abi + (bi)^2 - 1$
= $(a^2 - b^2 - 1) + 2abi$ (since $i^2 = -1$)
For $z_2 = a - bi$,
$z_2^2 - 1$
= $(a - bi)^2 - 1$
= $a^2 - 2abi + (bi)^2 - 1$
= $(a^2 - b^2 - 1) - 2abi$ (since $i^2 = -1$)