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

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