Find the difference quotient of $LaTeX: \displaystyle f(x)=- 3 x^{2} - 9 x + 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 3 \left(h + x\right)^{2} + 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 3 h^{2} - 6 h x - 9 h - 3 x^{2} - 9 x + 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 3 h^{2} - 6 h x - 9 h - 3 x^{2} - 9 x + 3)-(- 3 x^{2} - 9 x + 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 3 h^{2} - 6 h x - 9 h}{h}=- 3 h - 6 x - 9$