Question

The focus of a parabola is (−5,−1) and the directrix is y=−3.

Answer 1

$(x+5)^2=4(y+2)$

the equation for a parabola with focus (h, k+p) and directrix y=k-p is 
$(x-h)^2=4p(y-k)$

so using your directrix y=-3, and knowing directrix is y=k-p, you have 
-3=k-p.
similarly, knowing the focus is defined by (h,k+p) and your focus is (-5,-1)
you have the equation -1=k+p.

you now have a system of equations
$-3=k-p \\ -1=k+p$
which you can solve using any method, I will use elimination.
adding down
$-4=2k \\ -2=k$
you now have k and can find p using either equation. 
$-3=k-p \\ -3=-2-p \\ -1=-p \\ 1=p$.

now you plug those in, getting the answer
$(x+5)^2=4(y+2)$