Given the equation of the ellipse \(9x^2 + 4y^2 - 36x + 8y - 4 = 0\), find the vertex form.

Step 1 ：First, rewrite the ellipse equation with x terms and y terms grouped together: \(9x^2 - 36x + 4y^2 + 8y= 4\)

Step 2 ：Next, divide every term by the coefficient of \(x^2\) in the x-group and \(y^2\) in the y-group: \((x^2 - 4x) + (y^2 + 2y) = \frac{4}{9}\)

Step 3 ：Now, complete the square for the x-group and the y-group. To complete the square, take half of the coefficient of x, square it and add it to both sides. Do the same for y: \((x^2 - 4x + 4) + (y^2 + 2y + 1) = \frac{4}{9} + \frac{4}{9} + \frac{1}{4}\)

Step 4 ：The left-hand side of the equation can be rewritten as squares of binomials and the right-hand side can be calculated: \((x - 2)^2 + (y + 1)^2 = \frac{17}{36}\)