To find the vertex of the parabola represented by the function \( h = -6t^2 - 24t + 20 \), we can use the vertex formula for a quadratic equation in the form \( h = at^2 + bt + c \). The \( t \)-coordinate of the vertex is given by the formula \( t = -\frac{b}{2a} \).
In this equation:
- \( a = -6 \)
- \( b = -24 \)
Substituting in the values into the formula gives us:
\[ t = -\frac{-24}{2 \times -6} = \frac{24}{-12} = -2 \]
Now, to find the \( h \)-coordinate of the vertex, we substitute \( t = -2 \) back into the equation:
\[ h = -6(-2)^2 - 24(-2) + 20 \] \[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 44 \]
Thus, the vertex of the parabola occurs at the point \( (-2, 44) \).
The ordered pair that represents the vertex is:
\[ \boxed{(-2, 44)} \]