To solve the inequality \(4x^2 - 12x > 7\), follow these steps:

1. Rearrange the inequality:

\(4x^2 - 12x - 7 > 0\)

2. Factor the quadratic expression:

\(4x^2 - 12x - 7 = (2x - 7)(2x + 1)\)

3. Solve the inequality \((2x - 7)(2x + 1) > 0\):

- The critical points are \(x = \frac{7}{2}\) and \(x = -\frac{1}{2}\).

4. Test intervals around the critical points:

- For \(x < -\frac{1}{2}\), choose \(x = -1\):

\((2(-1) - 7)(2(-1) + 1) = (-9)(-1) = 9 > 0\)

- For \(-\frac{1}{2} < x < \frac{7}{2}\), choose \(x = 0\):

\((2(0) - 7)(2(0) + 1) = (-7)(1) = -7 < 0\)

- For \(x > \frac{7}{2}\), choose \(x = 4\):

\((2(4) - 7)(2(4) + 1) = (1)(9) = 9 > 0\)

5. The solution is:

\(x > \frac{7}{2} \text{ or } x < -\frac{1}{2}\)