possible \( 1<\quad \begin{array}{l}\text { Find the equation for the circle with center }(-4,-1) \text { and passing through } \\ (-5,5) \text {. } \\ \text { Write the equation for the circle. } \\ \square \text { (Simplify your answer.) }\end{array} \)

To find the equation of the circle, we use the standard form of a circle's equation, which is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. The center here is \((-4, -1)\), so we have: \[ (x + 4)^2 + (y + 1)^2 = r^2 \] Next, we need to calculate the radius \(r\) by finding the distance from the center to the point \((-5, 5)\). Using the distance formula: \[ r = \sqrt{((-5) - (-4))^2 + (5 - (-1))^2} = \sqrt{(-1)^2 + (6)^2} = \sqrt{1 + 36} = \sqrt{37} \] Substituting \(r^2\) into the equation gives us: \[ (x + 4)^2 + (y + 1)^2 = 37 \] And there you have it! The equation of the circle is: \[ (x + 4)^2 + (y + 1)^2 = 37 \]