To find the slope of the line, we start by calculating the first derivative of the function, as following:
[tex]\begin{gathered} f(x)=-0.5x^3-4x^2+2x-6 \\ \rightarrow f^{\prime}(x)=-1.5x^2-8x+2 \end{gathered}[/tex]The slope of the line tangent to the graph at (-6, -54) is:
[tex]f^{\prime}(-6)=m=-100[/tex]Then, we use the slope-point form to find the equation:
[tex]\begin{gathered} y+54=-100(x+6) \\ \rightarrow y+54=-100x-600 \\ \rightarrow y=-100x-654 \end{gathered}[/tex]