Find the difference quotient of $LaTeX: \displaystyle f(x)=- x^{2} - 3 x + 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 3 h - 3 x - \left(h + x\right)^{2} + 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=- h^{2} - 2 h x - 3 h - x^{2} - 3 x + 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- h^{2} - 2 h x - 3 h - x^{2} - 3 x + 8)-(- x^{2} - 3 x + 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- h^{2} - 2 h x - 3 h}{h}=- h - 2 x - 3$