(a) To express \(f(x) = -3x^2 + 12x + 2\) in the form \(-3(x-a)^2 + b\), we complete the square:

Start with \(-3(x^2 - 4x) + 2\).

Complete the square for \(x^2 - 4x\):

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

Substitute back:

\(-3((x-2)^2 - 4) + 2 = -3(x-2)^2 + 12 + 2 = -3(x-2)^2 + 14\).

Thus, \(a = 2\) and \(b = 14\).

(e) Given the translation vector \(\begin{pmatrix} -3 \\ 1 \end{pmatrix}\), the function \(g(x)\) is obtained by substituting \(x+3\) into \(f(x)\) and adding 1:

\(g(x) = -3((x+3)-2)^2 + 14 + 1\).

Simplify:

\(g(x) = -3(x+1)^2 + 15\).

Expand \(-3(x+1)^2\):

\(-3(x^2 + 2x + 1) = -3x^2 - 6x - 3\).

Thus, \(g(x) = -3x^2 - 6x - 3 + 15 = -3x^2 - 6x + 12\).