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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x - 4 \left(h + x\right)^{3} + 5 \left(h + x\right)^{2} + 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 4 h^{3} - 12 h^{2} x + 5 h^{2} - 12 h x^{2} + 10 h x - 4 h - 4 x^{3} + 5 x^{2} - 4 x + 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 4 h^{3} - 12 h^{2} x + 5 h^{2} - 12 h x^{2} + 10 h x - 4 h - 4 x^{3} + 5 x^{2} - 4 x + 4)-(- 4 x^{3} + 5 x^{2} - 4 x + 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 4 h^{3} - 12 h^{2} x + 5 h^{2} - 12 h x^{2} + 10 h x - 4 h}{h}=- 4 h^{2} - 12 h x + 5 h - 12 x^{2} + 10 x - 4$