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 .