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

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