Find the difference quotient of $LaTeX: \displaystyle f(x)=- 2 x^{2} + 9 x + 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x - 2 \left(h + x\right)^{2} + 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 2 h^{2} - 4 h x + 9 h - 2 x^{2} + 9 x + 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 2 h^{2} - 4 h x + 9 h - 2 x^{2} + 9 x + 4)-(- 2 x^{2} + 9 x + 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 2 h^{2} - 4 h x + 9 h}{h}=- 2 h - 4 x + 9$