(a) For a geometric progression, the ratio between consecutive terms is constant. Therefore, we have:

\(\frac{5p}{2p + 6} = \frac{8p + 2}{5p}\)

Cross-multiplying gives:

\(5p(5p) = (2p + 6)(8p + 2)\)

\(25p^2 = 16p^2 + 12p + 48p + 12\)

\(25p^2 = 16p^2 + 60p + 12\)

Rearranging gives:

\(9p^2 - 60p - 12 = 0\)

Factoring gives:

\((9p + 2)(p - 6) = 0\)

Thus, \(p = \frac{-2}{9}\) or \(p = 6\).

(b) Using \(p = \frac{-2}{9}\), the first term \(a\) is:

\(a = 2\left(\frac{-2}{9}\right) + 6 = \frac{50}{9}\)

The common ratio \(r\) is:

\(r = \frac{5p}{2p + 6} = \frac{10/9}{50/9} = \frac{-1}{5}\)

The sum to infinity \(S_\infty\) is given by:

\(S_\infty = \frac{a}{1 - r} = \frac{50/9}{1 - (-1/5)} = \frac{50/9}{6/5} = \frac{125}{27}\)