Start with the function \(y = h(x) = 4x^2 - 12x + 13\).

Complete the square for the quadratic expression:

\(4x^2 - 12x + 13 = 4(x^2 - 3x) + 13\).

\(= 4((x - \frac{3}{2})^2 - \frac{9}{4}) + 13\).

\(= 4(x - \frac{3}{2})^2 - 9 + 13\).

\(= 4(x - \frac{3}{2})^2 + 4\).

Set \(y = 4(x - \frac{3}{2})^2 + 4\).

Rearrange to solve for \(x\):

\(y - 4 = 4(x - \frac{3}{2})^2\).

\((x - \frac{3}{2})^2 = \frac{y - 4}{4}\).

\(x - \frac{3}{2} = \pm \sqrt{\frac{y - 4}{4}}\).

Since \(x < 0\), choose the negative root:

\(x = \frac{3}{2} - \frac{\sqrt{y - 4}}{2}\).

Thus, \(h^{-1}(x) = \frac{3}{2} - \frac{\sqrt{x - 4}}{2}\).