Find the difference quotient of $LaTeX: \displaystyle f(x)=8 x^{3} + x^{2} + 5 x + 5$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=5 h + 5 x + 8 \left(h + x\right)^{3} + \left(h + x\right)^{2} + 5$ and expanding gives $LaTeX: \displaystyle f(x+h)=8 h^{3} + 24 h^{2} x + h^{2} + 24 h x^{2} + 2 h x + 5 h + 8 x^{3} + x^{2} + 5 x + 5$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(8 h^{3} + 24 h^{2} x + h^{2} + 24 h x^{2} + 2 h x + 5 h + 8 x^{3} + x^{2} + 5 x + 5)-(8 x^{3} + x^{2} + 5 x + 5)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{8 h^{3} + 24 h^{2} x + h^{2} + 24 h x^{2} + 2 h x + 5 h}{h}=8 h^{2} + 24 h x + h + 24 x^{2} + 2 x + 5$