A) Find the focus, directrix, vertex and axis of symmetry for the parabola:
$$8(y+2)=(x+1{)}^2$$

Focus
Directrix
Vertex

Step 1 ：The given equation is in the form of $4p(y-k)=(x-h{)}^2$, which is the standard form of a parabola that opens upwards or downwards.

Step 2 ：Here, $h=-1$, $k=-2$ and $4p=8$, so $p=2$.

Step 3 ：The vertex of the parabola is at the point $(h,k)$, the focus is at the point $(h,k+p)$, and the directrix is the line $y=k-p$. The axis of symmetry is the vertical line $x=h$.

Step 4 ：Substituting the values of $h$, $k$, and $p$ into the formulas, we get the vertex at the point $(-1,-2)$, the focus at the point $(-1,0)$, the directrix at the line $y=-4$, and the axis of symmetry at the vertical line $x=-1$.

Step 5 ：$\boxed{{\displaystyle \text{Final Answer: The vertex of the parabola is at the point }(-1,-2),\text{ the focus is at the point }(-1,0),\text{ the directrix is the line }y=-4,\text{ and the axis of symmetry is the vertical line }x=-1}}$