Find the difference quotient of $LaTeX: \displaystyle f(x)=- 10 x^{3} - x^{2} + 8 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=8 h + 8 x - 10 \left(h + x\right)^{3} - \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 10 h^{3} - 30 h^{2} x - h^{2} - 30 h x^{2} - 2 h x + 8 h - 10 x^{3} - x^{2} + 8 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 10 h^{3} - 30 h^{2} x - h^{2} - 30 h x^{2} - 2 h x + 8 h - 10 x^{3} - x^{2} + 8 x + 9)-(- 10 x^{3} - x^{2} + 8 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 10 h^{3} - 30 h^{2} x - h^{2} - 30 h x^{2} - 2 h x + 8 h}{h}=- 10 h^{2} - 30 h x - h - 30 x^{2} - 2 x + 8$