Find the difference quotient of $LaTeX: \displaystyle f(x)=- 8 x^{3} + 2 x^{2} + 9 x + 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x - 8 \left(h + x\right)^{3} + 2 \left(h + x\right)^{2} + 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 8 h^{3} - 24 h^{2} x + 2 h^{2} - 24 h x^{2} + 4 h x + 9 h - 8 x^{3} + 2 x^{2} + 9 x + 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 8 h^{3} - 24 h^{2} x + 2 h^{2} - 24 h x^{2} + 4 h x + 9 h - 8 x^{3} + 2 x^{2} + 9 x + 5)-(- 8 x^{3} + 2 x^{2} + 9 x + 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 8 h^{3} - 24 h^{2} x + 2 h^{2} - 24 h x^{2} + 4 h x + 9 h}{h}=- 8 h^{2} - 24 h x + 2 h - 24 x^{2} + 4 x + 9$