(a) To express \(2x^2 - 8x + 14\) in the form \(2[(x-a)^2 + b]\), we complete the square for the quadratic expression:

Start with \(x^2 - 4x\). Complete the square:

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

Thus, \(2x^2 - 8x = 2((x-2)^2 - 4) = 2(x-2)^2 - 8\).

Now, add 14: \(2(x-2)^2 - 8 + 14 = 2(x-2)^2 + 6\).

Therefore, \(2x^2 - 8x + 14 = 2[(x-2)^2 + 3]\).

(b) To map \(y = f(x) = x^2\) to \(y = g(x) = 2x^2 - 8x + 14\), we perform the following transformations:

1. Translate by \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\): This shifts the graph right by 2 units and up by 3 units.

OR

1. Stretch in the \(y\) direction by a factor of 2: This scales the graph vertically.