Given the equation x^2 + 6x + 4y - 7 = 0 determine:

a) The Vertex V

b) If the parabola opens up, down, left, right

c) The Focus F

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First, solve the equation for y.

Add -4y to both sides.

x^2 + 6x + 4y - 7 + (-4y) = 0 + (-4y)

x^2 + 6x - 7 = -4y

Divide both sides by -4.

-0.25x^2 + -1.5x + 1.75 = y

Second, identify the values of a, b, and c.

y = ax^2 + bx + c

a = -0.25

b = -1.5

c = 1.75

Next, calculate the vertex's x-value using the formula:

x = -b/2a

x = -(-1.5) / 2 * (-0.25)

x = 1.5 / (-0.5)

x = -3

Now substiute -3 in for x and solve for y. This will give you the vertex's y-value.

(-0.25) * -3^2 + (-1.5) * -3 + 1.75 = y

(-0.25) * 9 + 4.5 + 1.75 = y

(-2.25) + 4.5 + 1.75 = y

y = 4

The vertex is (-3, 4).

Since the value of a is negative, the parabola opens down.

The focus of a parabola is a fixed point on the interior of the parabola. Since the parabola opens down, the x-value of the focus is the same as the x-value of the vertex. To find the y-value of the focus, use this formula:

y = c - (b^2 - 1)/4a

y = 1.75 - (-1.5^2 - 1) / 4 * -0.25

y = 1.75 - (2.25 - 1) / 4 * -0.25

y = 1.75 - 1.25 / -1

y = 1.75 + 1.25

y = 3

The focus is (-3, 3).

Here is a graph of the parabola:

**Summary:**

**a. Vertex = (-3, 4)**

**b. Parabola opens down**

**c. Focus = (-3, 3)**

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