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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=6 h + 6 x - 2 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} - 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 2 h^{3} - 6 h^{2} x + 6 h^{2} - 6 h x^{2} + 12 h x + 6 h - 2 x^{3} + 6 x^{2} + 6 x - 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 2 h^{3} - 6 h^{2} x + 6 h^{2} - 6 h x^{2} + 12 h x + 6 h - 2 x^{3} + 6 x^{2} + 6 x - 5)-(- 2 x^{3} + 6 x^{2} + 6 x - 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 2 h^{3} - 6 h^{2} x + 6 h^{2} - 6 h x^{2} + 12 h x + 6 h}{h}=- 2 h^{2} - 6 h x + 6 h - 6 x^{2} + 12 x + 6$