Find the difference quotient of $LaTeX: \displaystyle f(x)=- 9 x^{3} + 4 x^{2} - 5 x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 5 h - 5 x - 9 \left(h + x\right)^{3} + 4 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 9 h^{3} - 27 h^{2} x + 4 h^{2} - 27 h x^{2} + 8 h x - 5 h - 9 x^{3} + 4 x^{2} - 5 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 9 h^{3} - 27 h^{2} x + 4 h^{2} - 27 h x^{2} + 8 h x - 5 h - 9 x^{3} + 4 x^{2} - 5 x - 8)-(- 9 x^{3} + 4 x^{2} - 5 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 9 h^{3} - 27 h^{2} x + 4 h^{2} - 27 h x^{2} + 8 h x - 5 h}{h}=- 9 h^{2} - 27 h x + 4 h - 27 x^{2} + 8 x - 5$