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

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