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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x - 3 \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 3 h^{3} - 9 h^{2} x - 8 h^{2} - 9 h x^{2} - 16 h x - 4 h - 3 x^{3} - 8 x^{2} - 4 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 3 h^{3} - 9 h^{2} x - 8 h^{2} - 9 h x^{2} - 16 h x - 4 h - 3 x^{3} - 8 x^{2} - 4 x + 9)-(- 3 x^{3} - 8 x^{2} - 4 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 3 h^{3} - 9 h^{2} x - 8 h^{2} - 9 h x^{2} - 16 h x - 4 h}{h}=- 3 h^{2} - 9 h x - 8 h - 9 x^{2} - 16 x - 4$