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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 7 h - 7 x - 4 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 4 h^{2} - 8 h x - 7 h - 4 x^{2} - 7 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 4 h^{2} - 8 h x - 7 h - 4 x^{2} - 7 x - 7)-(- 4 x^{2} - 7 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 4 h^{2} - 8 h x - 7 h}{h}=- 4 h - 8 x - 7$