Find the difference quotient of $LaTeX: \displaystyle f(x)=- 8 x^{3} - 9 x^{2} + 7 x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=7 h + 7 x - 8 \left(h + x\right)^{3} - 9 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 8 h^{3} - 24 h^{2} x - 9 h^{2} - 24 h x^{2} - 18 h x + 7 h - 8 x^{3} - 9 x^{2} + 7 x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 8 h^{3} - 24 h^{2} x - 9 h^{2} - 24 h x^{2} - 18 h x + 7 h - 8 x^{3} - 9 x^{2} + 7 x - 3)-(- 8 x^{3} - 9 x^{2} + 7 x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 8 h^{3} - 24 h^{2} x - 9 h^{2} - 24 h x^{2} - 18 h x + 7 h}{h}=- 8 h^{2} - 24 h x - 9 h - 24 x^{2} - 18 x + 7$