Find and sketch through important points, i.e., x, y-intercept, vertex, etc.1. y^2 + 6y +8x +25=0 Please explain and show detailed step by step...

To sketch   y^2 + 6y +8x +25=0

We
know that this is second degree expression in y. So it is
parabola.

Now put this in he standard form (y-k)^2 = 4a(x-h) , which
is  astandard parabola with vertex at (h,k) and focal length a . The focus being located at (h+a
, k).

y^2+6y  =  -8x-25

We add 3^2 = 9
both sides:

y^2-6y +3^2 =
-8x-25+9

(y-3)^2 = - 8x-16.

(y-3)^2 = 
4(-2)(x+2).

Therefore this is a parabola  with vertex at (h,k) = (-2
, 3) with focus at (-2-2 , 3) = (-4, 3) and focal length of 2.

The
axis of symmetry is y = 3.

The parabola is open towards
left.

The parabola intercepts x axis  at x where (0-3)^2 = -8(x+2) .
Or at x= (9/-8) - 2 = -25/8. The parabola  has no y intercepts.intercepts
.