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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 2 h - 2 x - 10 \left(h + x\right)^{2} + 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 10 h^{2} - 20 h x - 2 h - 10 x^{2} - 2 x + 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 10 h^{2} - 20 h x - 2 h - 10 x^{2} - 2 x + 8)-(- 10 x^{2} - 2 x + 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 10 h^{2} - 20 h x - 2 h}{h}=- 10 h - 20 x - 2$