Find the difference quotient of $LaTeX: \displaystyle f(x)=3 x^{3} + 8 x^{2} + 3 x + 9$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=3 h + 3 x + 3 \left(h + x\right)^{3} + 8 \left(h + x\right)^{2} + 9$ and expanding gives $LaTeX: \displaystyle f(x+h)=3 h^{3} + 9 h^{2} x + 8 h^{2} + 9 h x^{2} + 16 h x + 3 h + 3 x^{3} + 8 x^{2} + 3 x + 9$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(3 h^{3} + 9 h^{2} x + 8 h^{2} + 9 h x^{2} + 16 h x + 3 h + 3 x^{3} + 8 x^{2} + 3 x + 9)-(3 x^{3} + 8 x^{2} + 3 x + 9)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{3 h^{3} + 9 h^{2} x + 8 h^{2} + 9 h x^{2} + 16 h x + 3 h}{h}=3 h^{2} + 9 h x + 8 h + 9 x^{2} + 16 x + 3$