$x^{2}-2 x+15< 0$

Step 1 ：Find the roots of the equation \(x^{2}-2 x+15=0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=15\).

Step 2 ：Substitute these values into the quadratic formula to get \(x = \frac{-(-2) \pm \sqrt{(-2)^{2}-4*1*15}}{2*1} = \frac{2 \pm \sqrt{4-60}}{2} = \frac{2 \pm \sqrt{-56}}{2}\).

Step 3 ：Since the term under the square root is negative, there are no real roots for the equation.

Step 4 ：Analyze the inequality. Since there are no real roots, the quadratic function \(x^{2}-2 x+15\) does not cross the x-axis. The coefficient of \(x^{2}\) is positive, which means the parabola opens upwards.

Step 5 ：Since the parabola opens upwards and does not cross the x-axis, the function \(x^{2}-2 x+15\) is always greater than 0 for all real values of x.

Step 6 ：Therefore, there are no real solutions to the inequality \(x^{2}-2 x+15<0\).

Step 7 ：Check the solution. Substituting any real number into the inequality \(x^{2}-2 x+15<0\) will result in a value greater than 0, confirming that there are no real solutions to the inequality.

Step 8 ：So, the solution to the inequality \(x^{2}-2 x+15<0\) is the empty set. \(\boxed{\text{No Solution}}\)