Find the difference quotient of $LaTeX: \displaystyle f(x)=- 4 x^{2} + 5 x + 6$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=5 h + 5 x - 4 \left(h + x\right)^{2} + 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=- 4 h^{2} - 8 h x + 5 h - 4 x^{2} + 5 x + 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(- 4 h^{2} - 8 h x + 5 h - 4 x^{2} + 5 x + 6)-(- 4 x^{2} + 5 x + 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{- 4 h^{2} - 8 h x + 5 h}{h}=- 4 h - 8 x + 5$