(a) To express \(x^2 - 8x + 11\) in the form \((x + p)^2 + q\), complete the square:

Start with \(x^2 - 8x\).

Take half of the coefficient of \(x\), square it, and add and subtract it inside the expression:

\(x^2 - 8x = (x - 4)^2 - 16\).

Now add 11 to balance the equation:

\((x - 4)^2 - 16 + 11 = (x - 4)^2 - 5\).

Thus, \(p = -4\) and \(q = -5\).

(b) Solve \((x - 4)^2 - 5 = 1\).

First, add 5 to both sides:

\((x - 4)^2 = 6\).

Take the square root of both sides:

\(x - 4 = \pm \sqrt{6}\).

Finally, solve for \(x\):

\(x = 4 \pm \sqrt{6}\).