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

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=h + x + \left(h + x\right)^{3} + 2 \left(h + x\right)^{2} - 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=h^{3} + 3 h^{2} x + 2 h^{2} + 3 h x^{2} + 4 h x + h + x^{3} + 2 x^{2} + x - 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(h^{3} + 3 h^{2} x + 2 h^{2} + 3 h x^{2} + 4 h x + h + x^{3} + 2 x^{2} + x - 6)-(x^{3} + 2 x^{2} + x - 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{h^{3} + 3 h^{2} x + 2 h^{2} + 3 h x^{2} + 4 h x + h}{h}=h^{2} + 3 h x + 2 h + 3 x^{2} + 4 x + 1$