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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 2 h - 2 x - \left(h + x\right)^{3} - 8 \left(h + x\right)^{2} + 10$ and expanding gives $LaTeX: \displaystyle f(x+h)=- h^{3} - 3 h^{2} x - 8 h^{2} - 3 h x^{2} - 16 h x - 2 h - x^{3} - 8 x^{2} - 2 x + 10$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- h^{3} - 3 h^{2} x - 8 h^{2} - 3 h x^{2} - 16 h x - 2 h - x^{3} - 8 x^{2} - 2 x + 10)-(- x^{3} - 8 x^{2} - 2 x + 10)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- h^{3} - 3 h^{2} x - 8 h^{2} - 3 h x^{2} - 16 h x - 2 h}{h}=- h^{2} - 3 h x - 8 h - 3 x^{2} - 16 x - 2$