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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=3 h + 3 x - 8 \left(h + x\right)^{3} - 3 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 8 h^{3} - 24 h^{2} x - 3 h^{2} - 24 h x^{2} - 6 h x + 3 h - 8 x^{3} - 3 x^{2} + 3 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 8 h^{3} - 24 h^{2} x - 3 h^{2} - 24 h x^{2} - 6 h x + 3 h - 8 x^{3} - 3 x^{2} + 3 x + 9)-(- 8 x^{3} - 3 x^{2} + 3 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 8 h^{3} - 24 h^{2} x - 3 h^{2} - 24 h x^{2} - 6 h x + 3 h}{h}=- 8 h^{2} - 24 h x - 3 h - 24 x^{2} - 6 x + 3$