Find the difference quotient of $LaTeX: \displaystyle f(x)=- 6 x^{3} - 10 x^{2} + 4 x - 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=4 h + 4 x - 6 \left(h + x\right)^{3} - 10 \left(h + x\right)^{2} - 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 6 h^{3} - 18 h^{2} x - 10 h^{2} - 18 h x^{2} - 20 h x + 4 h - 6 x^{3} - 10 x^{2} + 4 x - 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 6 h^{3} - 18 h^{2} x - 10 h^{2} - 18 h x^{2} - 20 h x + 4 h - 6 x^{3} - 10 x^{2} + 4 x - 5)-(- 6 x^{3} - 10 x^{2} + 4 x - 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 6 h^{3} - 18 h^{2} x - 10 h^{2} - 18 h x^{2} - 20 h x + 4 h}{h}=- 6 h^{2} - 18 h x - 10 h - 18 x^{2} - 20 x + 4$