Find the difference quotient of $LaTeX: \displaystyle f(x)=- 8 x^{3} - 4 x^{2} + 4 x - 5$ .

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