Find the difference quotient of $LaTeX: \displaystyle f(x)=x^{3} + 5 x^{2} - 8 x + 6$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=- 8 h - 8 x + \left(h + x\right)^{3} + 5 \left(h + x\right)^{2} + 6$ and expanding gives $LaTeX: \displaystyle f(x+h)=h^{3} + 3 h^{2} x + 5 h^{2} + 3 h x^{2} + 10 h x - 8 h + x^{3} + 5 x^{2} - 8 x + 6$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(h^{3} + 3 h^{2} x + 5 h^{2} + 3 h x^{2} + 10 h x - 8 h + x^{3} + 5 x^{2} - 8 x + 6)-(x^{3} + 5 x^{2} - 8 x + 6)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{h^{3} + 3 h^{2} x + 5 h^{2} + 3 h x^{2} + 10 h x - 8 h}{h}=h^{2} + 3 h x + 5 h + 3 x^{2} + 10 x - 8$