Okay, so ladies and gentlemen, what we have in this case is x squared plus y squared plus 4x minus 10y minus seven equals zero. They are asking us to find the center and radius of the circle and sketch the graph. To do this, we need to determine if the equation is in standard form. The standard form equation of a circle is (X - H)^2 + (Y - K)^2 = R^2, where (H, K) is the center and R is the radius. Looking at the equation, we can see that it does not match the standard form. We need to complete the square to get it in the correct format. To complete the square, we need to rewrite the equation with the X terms and Y terms grouped together. So we have x squared + 4x + y squared - 10y - seven equals zero. Now we can complete the square for each term. For the X terms, we take the coefficient of x (which is 4) and divide it by 2, then square it. So (4/2)^2 equals 4. For the Y terms, we take the coefficient of y (which is -10) and divide it by 2, then square it. So (-10/2)^2 equals 25. Now we add these values to the equation. We have x squared + 4x + 4 + y squared - 10y + 25 equals zero. By completing the square, we have turned the quadratic terms into perfect square trinomials. However, we need to make sure the equation is still balanced. If we added a value on one side, we need to add it on the other side as well. In this case, we added 4, so we also need to add it to the other side. We also need to add 7 to the other side to get...