a) x² + 6x + 4y + 5 = 0

Since the squared variable is x², the parabola has a
vertical axis of symmetry We get the equation in the 
standard form:

(x - h)² = 4p(y - k) 

and the vertex will be (h,k) and the distance from
the vertex to the focus and directrix will be |p|.
If p > 0, the parabola will open upward, and if p < 0,
the parabola will open downward:

x² + 6x + 4y + 5 = 0

Isolate x terms on the left:

x² + 6x = -4y - 5

1. Multiply coefficient of x by 1/2:  6· = 3 
2. Square that resul:                 3² = 9
3. Add that result to both sides of the equation: 

x² + 6x + 9 = -4y - 5 + 9

Factor the left side as (x + 3)(x + 3) or (x + 3)²
Combine number terms on the right

(x + 3)² = -4y + 4

Factor out the coefficient of y on the right

(x + 3)² = -4(y - 1)

Compare that to

(x - h)² = 4p(y - k)

-h = +3, so h = -3 
-k = -1, so k = 1
vertex = (h,k) = (-3,1)
4p = -4, so p = -1
p is negative so parabola opens downward:
distance from vertex to focus = distance from vertex to directrix = 
|p| = |-1| = 1
Since the parabola opens downward, the focus is the point which is
|p| = 1 unit below the vertex at (-3,1-1) or (-3,0), and the directrix is
the horizontal line 1 units above the vertex or y = 1+1 or y = 2

We plot the vertex (-3,1), the focus (-3,0) and the directrix y = 2:

 


We plot the "latus rectum" or "focal chord", a horizontal line
4|p| or 4(1) or 4 units long bisected by the focus:

 

And sketch in the parabola:

 

--------------------------

b) y² + 6y - 8x - 31 = 0 

Since the squared variable is y², the parabola has a
horizontal axis of symmetry. We get the equation in the 
standard form:

(y - k)² = 4p(x - h) 

and the vertex will be (h,k) and the distance from
the vertex to the focus and directrix will be |p|.
If p > 0, the parabola will open to the right, and 
if p < 0, the parabola will open to the left:

y² + 6y - 8x - 31 = 0 

Isolate y terms on the left:

y² + 6y = 8x + 31 

1. Multiply coefficient of y by 1/2:  6· = 3 
2. Square that resul:                 3² = 9
3. Add that result to both sides of the equation: 

y² + 6y + 9 = 8x + 31 + 9

Factor the left side as (y + 3)(y + 3) or (y + 3)²
Combine number terms on the right

(y + 3)² = 8x + 40

Factor out the coefficient of y on the right

(y + 3)² = 8(x - 5)

Compare that to

(y - k)² = 4p(x - h)

-k = +3, so k = -3 
-h = -5, so h = 5

vertex = (h,k) = (5,-3)
4p = 8, so p = 2
p is positive so parabola opens to the right:
distance from vertex to focus = distance from vertex to directrix = 
|p| = |2| = 2
Since the parabola opens to the right, the focus is the point 
which is |p| = 2 unit to the right of the vertex at (5+2,-3) or 
(7,-3), and the directrix is the vertical line 2 units left of 
the vertex or x = 5-2 or x = 3

We plot the vertex (5,-3), the focus (7,-3) and the directrix x = 3:

 


We plot the "latus rectum" or "focal chord", a vertical line
4|p| or 4(1) or 4 units long bisected by the focus:

 

And sketch in the parabola:

 

Edwin