To find the vertex of the quadratic function \( h(t) = -6t^2 - 24t + 20 \), we can use the vertex formula for a quadratic in the form \( ax^2 + bx + c \). The \( t \)-coordinate of the vertex can be found using the formula:
\[ t = -\frac{b}{2a} \]
In this case, \( a = -6 \) and \( b = -24 \). Plugging in these values:
\[ t = -\frac{-24}{2 \times -6} = \frac{24}{-12} = -2 \]
Next, we can determine the \( h \)-coordinate of the vertex by substituting \( t = -2 \) back into the function:
\[ h(-2) = -6(-2)^2 - 24(-2) + 20 \] \[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 44 \]
So the vertex is at the point \( (-2, 44) \).
From the options provided, the ordered pair that represents the vertex is:
\[ (-2, 44) \]
Therefore, the correct answer is \( (-2, 44) \).