Try to use the circle formula!

For what value of c will the circle with equation \(x^2 + 6x + y^2 - 4y + c = 0\) have a radius of length 4?

x^2 + 6x + y^2 - 4y = -c complete the square on x and y

x^2 - 6x + 9 + y^2 - 4y + 4 = -c + 9 + 4

Note that if the radius is 4.....then the right side must = r^2 = 4^2 = 16

So

-c + 9 + 4 =16

-c + 13 =16

-c = 3

c = -3