x² + 2x + y2 - 6y =-6

Let's solve the equation step by step and rewrite it in a way that makes it easy to identify its geometric representation. The given equation is:

x² + 2x + y² - 6y = -6

To solve this equation, we need to complete the square for both the x-terms and the y-terms. This process will allow us to rewrite the equation in the form of a circle equation.

First, let's complete the square for the x-terms:

x² + 2x needs the addition of (2/2)² = 1² = 1 to complete the square.

So we add 1 to both sides:

x² + 2x + 1 + y² - 6y = -6 + 1

Now, the equation becomes:

(x + 1)² + y² - 6y = -5

Next, we complete the square for the y-terms:

y² - 6y needs the addition of (6/2)² = 3² = 9 to complete the square.

So we add 9 to both sides:

(x + 1)² + y² - 6y + 9 = -5 + 9

This gives us:

(x + 1)² + (y - 3)² = 4

Now the equation is in the standard form of a circle:

(x - h)² + (y - k)² = r²

Where (h,k) is the center of the circle and r is the radius.

Comparing, we have h = -1, k = 3, and r² = 4. Thus the radius r is √4 = 2.

The equation represents a circle with center at (-1, 3) and radius 2.