Answer
$a_2=-6$
and
$a_3=18$
Work Step by Step
Given: Geometric sequence: $2,a_2,a_3,-54$
Here $a_4=a_1r^{n-1}=a_1r^3$
Thus, $-54=2r^3 \implies r=-3$
Now, $a_2=a_1r^{2-1}=a_1r \implies (2)(-3)=-6$
and
$a_3=a_1r^{3-1}=a_1r^2 \implies (2)(9)=18$