Find the difference quotient of $LaTeX: \displaystyle f(x)=4 x^{2} - 5 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 5 h - 5 x + 4 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=4 h^{2} + 8 h x - 5 h + 4 x^{2} - 5 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(4 h^{2} + 8 h x - 5 h + 4 x^{2} - 5 x + 9)-(4 x^{2} - 5 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{4 h^{2} + 8 h x - 5 h}{h}=4 h + 8 x - 5$