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

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