(i) Since the roots of the equation are \(-3\) and \(5\), we can express the quadratic as \((x + 3)(x - 5) = 0\).

Expanding this, we get:

\(x^2 - 5x + 3x - 15 = x^2 - 2x - 15\).

Thus, \(p = -2\) and \(q = -15\).

(ii) For the equation \(x^2 + px + q + r = 0\) to have equal roots, the discriminant must be zero.

The discriminant \(b^2 - 4ac = 0\).

Here, \(a = 1\), \(b = p = -2\), and \(c = q + r = -15 + r\).

So, \((-2)^2 - 4(1)(-15 + r) = 0\).

\(4 - 4(-15 + r) = 0\).

\(4 + 60 - 4r = 0\).

\(64 = 4r\).

\(r = 16\).