To solve the inequality \(|x + 3a| > 2|x - 2a|\), we consider the non-modular inequality \((x + 3a)^2 > (2(x - 2a))^2\).

Expanding both sides, we have:

\((x + 3a)^2 = x^2 + 6ax + 9a^2\)

\((2(x - 2a))^2 = 4(x^2 - 4ax + 4a^2) = 4x^2 - 16ax + 16a^2\)

Thus, the inequality becomes:

\(x^2 + 6ax + 9a^2 > 4x^2 - 16ax + 16a^2\)

Rearranging terms gives:

\(-3x^2 + 22ax - 7a^2 > 0\)

Solving the quadratic equation \(-3x^2 + 22ax - 7a^2 = 0\), we find the critical values:

\(x = \frac{1}{3}a\) and \(x = 7a\).

Therefore, the solution to the inequality is:

\(\frac{1}{3}a < x < 7a\).