Find the difference quotient of $LaTeX: \displaystyle f(x)=- 10 x^{2} + 9 x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x - 10 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 10 h^{2} - 20 h x + 9 h - 10 x^{2} + 9 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 10 h^{2} - 20 h x + 9 h - 10 x^{2} + 9 x - 8)-(- 10 x^{2} + 9 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 10 h^{2} - 20 h x + 9 h}{h}=- 10 h - 20 x + 9$