Find the difference quotient of $LaTeX: \displaystyle f(x)=- 7 x^{2} - 4 x + 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x - 7 \left(h + x\right)^{2} + 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 7 h^{2} - 14 h x - 4 h - 7 x^{2} - 4 x + 8)-(- 7 x^{2} - 4 x + 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 7 h^{2} - 14 h x - 4 h}{h}=- 7 h - 14 x - 4$