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

Expanding both sides, we have:

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

\((3(x - a))^2 = 9(x^2 - 2ax + a^2) = 9x^2 - 18ax + 9a^2\)

Setting up the inequality:

\(x^2 + 4ax + 4a^2 > 9x^2 - 18ax + 9a^2\)

Simplifying, we get:

\(-8x^2 + 22ax - 5a^2 > 0\)

Solving the quadratic equation \(-8x^2 + 22ax - 5a^2 = 0\) using the quadratic formula:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

Here, \(a = -8\), \(b = 22a\), \(c = -5a^2\).

\(x = \frac{-22a \pm \sqrt{(22a)^2 - 4(-8)(-5a^2)}}{2(-8)}\)

\(x = \frac{-22a \pm \sqrt{484a^2 - 160a^2}}{-16}\)

\(x = \frac{-22a \pm \sqrt{324a^2}}{-16}\)

\(x = \frac{-22a \pm 18a}{-16}\)

\(x = \frac{-22a + 18a}{-16}\) or \(x = \frac{-22a - 18a}{-16}\)

\(x = \frac{-4a}{-16} = \frac{1}{4}a\) or \(x = \frac{-40a}{-16} = \frac{5}{2}a\)

Thus, the solution to the inequality is \(\frac{1}{4}a < x < \frac{5}{2}a\).