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

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