Final answer:
The center of the circle is (-3, -9/2) and the radius is sqrt(41)/2.
Step-by-step explanation:
To find the coordinates of the center and length of the radius of the circle, we need to rewrite the equation of the circle in the standard form (x-h)^2 + (y-k)^2 = r^2. First, complete the square for the x and y terms by adding and subtracting the necessary constants:
x^2 + 6x + 4y = 23
x^2 + 6x + 9 + 4y + 9 = 23 + 9 + 9
(x + 3)^2 + (y + 9/2) = 41/4
Therefore, the center of the circle is (-3, -9/2) and the radius is sqrt(41/4), which simplifies to sqrt(41)/2.