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

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