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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=6 h + 6 x - 3 \left(h + x\right)^{3} + 5 \left(h + x\right)^{2} + 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 3 h^{3} - 9 h^{2} x + 5 h^{2} - 9 h x^{2} + 10 h x + 6 h - 3 x^{3} + 5 x^{2} + 6 x + 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 3 h^{3} - 9 h^{2} x + 5 h^{2} - 9 h x^{2} + 10 h x + 6 h - 3 x^{3} + 5 x^{2} + 6 x + 4)-(- 3 x^{3} + 5 x^{2} + 6 x + 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 3 h^{3} - 9 h^{2} x + 5 h^{2} - 9 h x^{2} + 10 h x + 6 h}{h}=- 3 h^{2} - 9 h x + 5 h - 9 x^{2} + 10 x + 6$