Find the difference quotient of $LaTeX: \displaystyle f(x)=- 3 x^{2} - 4 x - 1$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x - 3 \left(h + x\right)^{2} - 1$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 3 h^{2} - 6 h x - 4 h - 3 x^{2} - 4 x - 1$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 3 h^{2} - 6 h x - 4 h - 3 x^{2} - 4 x - 1)-(- 3 x^{2} - 4 x - 1)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 3 h^{2} - 6 h x - 4 h}{h}=- 3 h - 6 x - 4$