Find the difference quotient of $LaTeX: \displaystyle f(x)=9 x^{3} + 6 x^{2} + 8 x - 4$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=8 h + 8 x + 9 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} - 4$ and expanding gives $LaTeX: \displaystyle f(x+h)=9 h^{3} + 27 h^{2} x + 6 h^{2} + 27 h x^{2} + 12 h x + 8 h + 9 x^{3} + 6 x^{2} + 8 x - 4$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(9 h^{3} + 27 h^{2} x + 6 h^{2} + 27 h x^{2} + 12 h x + 8 h + 9 x^{3} + 6 x^{2} + 8 x - 4)-(9 x^{3} + 6 x^{2} + 8 x - 4)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{9 h^{3} + 27 h^{2} x + 6 h^{2} + 27 h x^{2} + 12 h x + 8 h}{h}=9 h^{2} + 27 h x + 6 h + 27 x^{2} + 12 x + 8$