Find the difference quotient of $LaTeX: \displaystyle f(x)=- 4 x^{3} + x^{2} - 9 x + 10$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 9 h - 9 x - 4 \left(h + x\right)^{3} + \left(h + x\right)^{2} + 10$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 4 h^{3} - 12 h^{2} x + h^{2} - 12 h x^{2} + 2 h x - 9 h - 4 x^{3} + x^{2} - 9 x + 10$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 4 h^{3} - 12 h^{2} x + h^{2} - 12 h x^{2} + 2 h x - 9 h - 4 x^{3} + x^{2} - 9 x + 10)-(- 4 x^{3} + x^{2} - 9 x + 10)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 4 h^{3} - 12 h^{2} x + h^{2} - 12 h x^{2} + 2 h x - 9 h}{h}=- 4 h^{2} - 12 h x + h - 12 x^{2} + 2 x - 9$