Find the difference quotient of $LaTeX: \displaystyle f(x)=5 x^{2} - 8 x + 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 8 h - 8 x + 5 \left(h + x\right)^{2} + 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=5 h^{2} + 10 h x - 8 h + 5 x^{2} - 8 x + 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(5 h^{2} + 10 h x - 8 h + 5 x^{2} - 8 x + 5)-(5 x^{2} - 8 x + 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{5 h^{2} + 10 h x - 8 h}{h}=5 h + 10 x - 8$