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

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