Determine the vertex of the quadratic equation: 10) $y=-x^{2}+2 x+4$

Step 1 ：The vertex of a quadratic equation in the form \(y = ax^2 + bx + c\) is given by the point \(-\frac{b}{2a}, f(-\frac{b}{2a})\).

Step 2 ：For the given equation \(y = -x^2 + 2x + 4\), we have \(a = -1\) and \(b = 2\).

Step 3 ：So, the x-coordinate of the vertex is \(-\frac{b}{2a} = -\frac{2}{2*(-1)} = 1\).

Step 4 ：Substitute \(x = 1\) into the equation to find the y-coordinate of the vertex: \(y = -(1)^2 + 2*1 + 4 = -1 + 2 + 4 = 5\).

Step 5 ：So, the vertex of the quadratic equation \(y = -x^2 + 2x + 4\) is \((1, 5)\).

Step 6 ：\(\boxed{(1, 5)}\) is the final answer.