Step 1:
Vertex $(3,-2)$
Open downward and length of latus rectum is 8.
Latus rectum=$4a=8$
$\Rightarrow a=2$
Step 2:
The parabola opens downwards,the equation of the form $X^2=-4aY$-----(1)
Where $X=(x-3),Y=(y+2),a=2$
Substitute these in eq(1) we get
$(x-3)^2=-4\times 2\times (y+2)$
$\Rightarrow -8(y+2)$
$(x-3)^2=-8(y+2)$
This is the required equation.