Find the difference quotient of $LaTeX: \displaystyle f(x)=- 9 x^{3} + 6 x^{2} + 3 x + 3$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=3 h + 3 x - 9 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} + 3$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 9 h^{3} - 27 h^{2} x + 6 h^{2} - 27 h x^{2} + 12 h x + 3 h - 9 x^{3} + 6 x^{2} + 3 x + 3$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 9 h^{3} - 27 h^{2} x + 6 h^{2} - 27 h x^{2} + 12 h x + 3 h - 9 x^{3} + 6 x^{2} + 3 x + 3)-(- 9 x^{3} + 6 x^{2} + 3 x + 3)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 9 h^{3} - 27 h^{2} x + 6 h^{2} - 27 h x^{2} + 12 h x + 3 h}{h}=- 9 h^{2} - 27 h x + 6 h - 27 x^{2} + 12 x + 3$