(i) Start with the given equation:

\(2\ln(4x - 5) + \ln(x + 1) = 3\ln 3\)

Using the properties of logarithms, combine the logs:

\(\ln((4x - 5)^2) + \ln(x + 1) = \ln(27)\)

\(\ln((4x - 5)^2(x + 1)) = \ln(27)\)

Equating the arguments:

\((4x - 5)^2(x + 1) = 27\)

Expanding and simplifying gives:

\(16x^3 - 24x^2 - 15x - 2 = 0\)

(ii) Use the factor theorem to find a root. Testing \(x = 2\):

\(16(2)^3 - 24(2)^2 - 15(2) - 2 = 0\)

So \(x = 2\) is a root, and \((x - 2)\) is a factor.

Divide \(16x^3 - 24x^2 - 15x - 2\) by \((x - 2)\) to find the quotient:

The quotient is \(16x^2 + 8x + 1\).

Factorise further:

\((x - 2)(4x + 1)^2\)

(iii) Solve \(2\ln(4x - 5) + \ln(x + 1) = 3\ln 3\):

From part (i), \((4x - 5)^2(x + 1) = 27\).

Using the factorisation, \(x = 2\) is the only solution.