Reflect the point $(1,5)$ across the line $y=2 x+1$ to obtain the point $p$. Determine the coordinates for $p$.

Step 1 ：Plug the values into the formula:

Step 2 ：\(x' = \frac{1 - 2(5 - 1)}{1 + 2^2} = -\frac{7}{5}\)

Step 3 ：\(y' = \frac{2(1 - 1) + 5(1 - 2^2) + 2(1)}{1 + 2^2} = -\frac{11}{5}\)

Step 4 ：The coordinates of the reflected point are \(\boxed{\left(-\frac{7}{5}, -\frac{11}{5}\right)}\)