(a)(i) The expression \(\frac{4x}{(x+4)(x^2+3)}\) can be decomposed into partial fractions as \(\frac{A}{x+4} + \frac{Bx+C}{x^2+3}\) because \(x^2+3\) is an irreducible quadratic factor.

(a)(ii) The expression \(\frac{2x+1}{(x-2)(x+2)^2}\) can be decomposed into partial fractions as \(\frac{A}{x-2} + \frac{Bx+C}{(x+2)^2}\) because \((x+2)^2\) is a repeated linear factor.

(b) To show \(\int_3^4 \frac{3x}{(x+1)(x-2)} \, dx = \ln 5\), first express \(\frac{3x}{(x+1)(x-2)}\) as \(\frac{A}{x+1} + \frac{B}{x-2}\). Solving for \(A\) and \(B\), we find \(A = 1\) and \(B = 2\).

Integrate: \(\int \left( \frac{1}{x+1} + \frac{2}{x-2} \right) \, dx = \ln|x+1| + 2\ln|x-2| + C\).

Evaluate from 3 to 4: \(\left[ \ln|x+1| + 2\ln|x-2| \right]_3^4 = (\ln 5 + 2\ln 2) - (\ln 4 + 2\ln 1) = \ln 5\).