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

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