Find the difference quotient of $LaTeX: \displaystyle f(x)=- 2 x^{2} - 7 x - 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 7 h - 7 x - 2 \left(h + x\right)^{2} - 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 2 h^{2} - 4 h x - 7 h - 2 x^{2} - 7 x - 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 2 h^{2} - 4 h x - 7 h - 2 x^{2} - 7 x - 4)-(- 2 x^{2} - 7 x - 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 2 h^{2} - 4 h x - 7 h}{h}=- 2 h - 4 x - 7$