Find the difference quotient of $LaTeX: \displaystyle f(x)=- 8 x^{2} - 3 x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 3 h - 3 x - 8 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 8 h^{2} - 16 h x - 3 h - 8 x^{2} - 3 x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 8 h^{2} - 16 h x - 3 h - 8 x^{2} - 3 x - 3)-(- 8 x^{2} - 3 x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 8 h^{2} - 16 h x - 3 h}{h}=- 8 h - 16 x - 3$