Find the difference quotient of $LaTeX: \displaystyle f(x)=- x^{3} + 2 x^{2} + 7 x - 7$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=7 h + 7 x - \left(h + x\right)^{3} + 2 \left(h + x\right)^{2} - 7$ and expanding gives $LaTeX: \displaystyle f(x+h)=- h^{3} - 3 h^{2} x + 2 h^{2} - 3 h x^{2} + 4 h x + 7 h - x^{3} + 2 x^{2} + 7 x - 7$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- h^{3} - 3 h^{2} x + 2 h^{2} - 3 h x^{2} + 4 h x + 7 h - x^{3} + 2 x^{2} + 7 x - 7)-(- x^{3} + 2 x^{2} + 7 x - 7)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- h^{3} - 3 h^{2} x + 2 h^{2} - 3 h x^{2} + 4 h x + 7 h}{h}=- h^{2} - 3 h x + 2 h - 3 x^{2} + 4 x + 7$