(a) Express \(2x^2 + 12x + 11\) in the form \(2(x + a)^2 + b\), where \(a\) and \(b\) are constants.

The function \(f\) is defined by \(f(x) = 2x^2 + 12x + 11\) for \(x \leq -4\).

(b) Find an expression for \(f^{-1}(x)\) and state the domain of \(f^{-1}\).

The function \(g\) is defined by \(g(x) = 2x - 3\) for \(x \leq k\).

(c) For the case where \(k = -1\), solve the equation \(fg(x) = 193\).

(d) State the largest value of \(k\) possible for the composition \(fg\) to be defined.