(i) To express \(y = x^2 + 4x + 3\) in the form \(y = (x + a)^2 + b\), complete the square:

\(y = (x + 2)^2 - 1\).

Hence, \(a = 2\) and \(b = -1\).

To express \(x\) in terms of \(y\):

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

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

\(x + 2 = \pm \sqrt{y + 1}\)

Since \(x \geq -2\), we take the positive root:

\(x = -2 + \sqrt{y + 1}\).

(ii) To find the volume when the shaded region is rotated about the y-axis, use the formula for the volume of revolution:

\(V = \pi \int_{-1}^{3} x^2 \, dy\)

Substitute \(x = -2 + \sqrt{y + 1}\):

\(x^2 = 4 + (y + 1) - 4(y + 1)^{1/2}\)

\(V = \pi \int_{-1}^{3} \left[ 5y + \frac{y^2}{2} - \frac{4(y + 1)^{3/2}}{3/2} \right] \, dy\)

Integrate and apply limits:

\(V = \pi \left[ 15 + \frac{9}{2} - \frac{64}{3} - \left(-5 + \frac{1}{2}\right) \right]\)

\(V = \frac{8\pi}{3}\) or 8.38