Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{3} - 4 x^{2} - 3 x - 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 3 h - 3 x - 6 \left(h + x\right)^{3} - 4 \left(h + x\right)^{2} - 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{3} - 18 h^{2} x - 4 h^{2} - 18 h x^{2} - 8 h x - 3 h - 6 x^{3} - 4 x^{2} - 3 x - 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{3} - 18 h^{2} x - 4 h^{2} - 18 h x^{2} - 8 h x - 3 h - 6 x^{3} - 4 x^{2} - 3 x - 3)-(- 6 x^{3} - 4 x^{2} - 3 x - 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{3} - 18 h^{2} x - 4 h^{2} - 18 h x^{2} - 8 h x - 3 h}{h}=- 6 h^{2} - 18 h x - 4 h - 18 x^{2} - 8 x - 3$