Find the difference quotient of $LaTeX: \displaystyle f(x)=- 7 x^{3} - 10 x^{2} - 9 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 7 \left(h + x\right)^{3} - 10 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{3} - 21 h^{2} x - 10 h^{2} - 21 h x^{2} - 20 h x - 9 h - 7 x^{3} - 10 x^{2} - 9 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{3} - 21 h^{2} x - 10 h^{2} - 21 h x^{2} - 20 h x - 9 h - 7 x^{3} - 10 x^{2} - 9 x + 9)-(- 7 x^{3} - 10 x^{2} - 9 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{3} - 21 h^{2} x - 10 h^{2} - 21 h x^{2} - 20 h x - 9 h}{h}=- 7 h^{2} - 21 h x - 10 h - 21 x^{2} - 20 x - 9$