First, find the coefficient of \(x^4\) in \((3 + x)^5\). Using the binomial theorem, the general term is \(\binom{5}{r} 3^{5-r} x^r\). For \(x^4\), set \(r = 4\):

\(\binom{5}{4} 3^{1} x^4 = 5 \times 3 = 15\).

Next, find the coefficient of \(x^2\) in \(\left(2x + \frac{a}{x}\right)^6\). The general term is \(\binom{6}{r} (2x)^{6-r} \left(\frac{a}{x}\right)^r\).

For \(x^2\), solve \(6-r-r = 2\), giving \(r = 2\).

The term is \(\binom{6}{2} (2x)^{4} \left(\frac{a}{x}\right)^2 = 15 \times 16x^4 \times \frac{a^2}{x^2} = 240a^2 x^2\).

Equating the coefficients: \(15 = 240a^2\).

Solving for \(a\):

\(a^2 = \frac{15}{240} = \frac{1}{16}\).

\(a = \frac{1}{4}\) (since \(a\) is positive).