(a) To translate the curve \(y = x^2 + 2x - 5\) by \(\begin{pmatrix} -1 \\ 3 \end{pmatrix}\), we substitute \(x + 1\) for \(x\) in the equation and add 3 to the result:

\(y = (x+1)^2 + 2(x+1) - 5 + 3\)

Expanding, we get:

\(y = (x^2 + 2x + 1) + (2x + 2) - 5 + 3\)

\(y = x^2 + 4x + 1\)

(b) The transformation from \(y = x^2 + 2x - 5\) to \(y = 4x^2 + 4x - 5\) is a horizontal stretch. The coefficient of \(x^2\) changes from 1 to 4, indicating a stretch in the x-direction by a factor of \(\frac{1}{2}\), keeping the y-axis invariant.