Find the difference quotient of $LaTeX: \displaystyle f(x)=- 7 x^{2} - 9 x - 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 7 \left(h + x\right)^{2} - 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{2} - 14 h x - 9 h - 7 x^{2} - 9 x - 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{2} - 14 h x - 9 h - 7 x^{2} - 9 x - 5)-(- 7 x^{2} - 9 x - 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{2} - 14 h x - 9 h}{h}=- 7 h - 14 x - 9$