(a) To express \(f(x) = x^2 - 4x + 9\) in the form \((x-a)^2 + b\), complete the square:

\(f(x) = (x-2)^2 + 5\)

(b) To express \(g(x) = 2x^2 + 4x + 12\) in the form \(2[(x+c)^2 + d]\), first factor out the 2:

\(g(x) = 2(x^2 + 2x + 6)\)

Complete the square for \(x^2 + 2x + 6\):

\(x^2 + 2x + 6 = (x+1)^2 + 5\)

Thus, \(g(x) = 2[(x+1)^2 + 5]\)

(c) Express \(g(x)\) in the form \(kf(x+h)\):

\(g(x) = 2(x^2 + 2x + 6) = 2[(x+3)^2 + 5]\)

So, \(g(x) = 2f(x+3)\) where \(k=2\) and \(h=3\)

(d) The transformations from \(y = f(x)\) to \(y = g(x)\) are:

- Translation by \(\begin{pmatrix} -3 \\ 0 \end{pmatrix}\)

- Stretch in the y-direction by a factor of 2