identify the conic section. if its a parabols give the vertex if its a circle, give the center and radius. if it is an ellipse or a hyperbola, give the center and foci.

again my teacher said the answer is ellipse with center (-4,4), foci at (-4± V3, 4) but don't get how she got it

You have not given us the problem that we are supposed to solve. I can not even reverse engineer it from what you have written. What conic section?

sorry the problem is 4x^2 + 7y^2 + 32x-56y +148=0

4 x^2 + 32 x + 7 y^2 -56 y = -148

x^2 + 8 x + (7/4)y^2 - 14 y = -37
x^2 + 8 x + 16 + (7/4) y^2 -14 y = -21
(x+4)^2 + (7/4) y^2 - 14 y = -21
(4/7)(x+4)^2 + y^2 - 8 y = -12
(4/7)(x+4)^2 + y^2-8y+16 = 4
(4/7)(x+4)^2 + (y-4)^2 = 4
(x+4)^2 /7 + (y-4)^2/4 = 1
ellipse with center at (-4,4)
a^2 = 7
b^2 4
so a = sqrt 7
b = 2
center to focus = sqrt(7-4) = sqrt 3
etc

To identify the conic section and determine its center, foci, or vertex, we need to analyze the given information. In this case, your teacher provided the conic section as an ellipse with the center at (-4,4) and foci at (-4±√3,4). Let's break down the process.

1. The center of the ellipse:
The given center is (-4,4). The center represents the midpoint of the major axis (longest diameter) and the minor axis (shortest diameter) of the ellipse.

2. The foci of the ellipse:
The foci are defined as two fixed points inside the ellipse that play a crucial role in determining its shape. In this case, the foci are located at (-4±√3,4).

Now, let's try to understand how your teacher obtained these values.

- For an ellipse, the equation in general form is:
((x-h)^2)/a^2 + ((y-k)^2)/b^2 = 1,
where (h,k) is the center of the ellipse, 'a' represents the distance from the center to the vertices along the x-axis, and 'b' represents the distance from the center to the vertices along the y-axis.

- In an ellipse, the distance between the center and each focus (c) is related to the values of 'a' and 'b' by the equation:
c = √(a^2 - b^2).

- The value of 'a' represents the semi-major axis (half of the major axis) of the ellipse, while 'b' represents the semi-minor axis (half of the minor axis).

Based on the information given by your teacher, the distance between the center (-4,4) and each focus should be √3. Therefore, we can set up the following equation:

c = √(a^2 - b^2),
√3 = √(a^2 - b^2).

Since the center of the ellipse is given as (-4,4), we can assume that the x-coordinate of each focus is -4. Therefore, we have:

c = √(a^2 - b^2),
√3 = √((-4 - (-4))^2 - b^2),
√3 = √(0^2 - b^2),
√3 = √(-b^2),
3 = -b^2.

As the equation has no real solutions, it appears that there may be an error in the information provided by your teacher. It is not possible to have an ellipse with foci at (-4±√3,4) if the center is located at (-4,4). Please validate the information before proceeding.