Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{2} + 9 x - 2$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=9 h + 9 x - 6 \left(h + x\right)^{2} - 2$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{2} - 12 h x + 9 h - 6 x^{2} + 9 x - 2$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{2} - 12 h x + 9 h - 6 x^{2} + 9 x - 2)-(- 6 x^{2} + 9 x - 2)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{2} - 12 h x + 9 h}{h}=- 6 h - 12 x + 9$