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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 4 h - 4 x + 6 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} + 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=6 h^{3} + 18 h^{2} x + 6 h^{2} + 18 h x^{2} + 12 h x - 4 h + 6 x^{3} + 6 x^{2} - 4 x + 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(6 h^{3} + 18 h^{2} x + 6 h^{2} + 18 h x^{2} + 12 h x - 4 h + 6 x^{3} + 6 x^{2} - 4 x + 8)-(6 x^{3} + 6 x^{2} - 4 x + 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{6 h^{3} + 18 h^{2} x + 6 h^{2} + 18 h x^{2} + 12 h x - 4 h}{h}=6 h^{2} + 18 h x + 6 h + 18 x^{2} + 12 x - 4$