Find the difference quotient of $LaTeX: \displaystyle f(x)=10 x^{3} + 6 x^{2} + 6 x - 2$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=6 h + 6 x + 10 \left(h + x\right)^{3} + 6 \left(h + x\right)^{2} - 2$ and expanding gives $LaTeX: \displaystyle f(x+h)=10 h^{3} + 30 h^{2} x + 6 h^{2} + 30 h x^{2} + 12 h x + 6 h + 10 x^{3} + 6 x^{2} + 6 x - 2$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(10 h^{3} + 30 h^{2} x + 6 h^{2} + 30 h x^{2} + 12 h x + 6 h + 10 x^{3} + 6 x^{2} + 6 x - 2)-(10 x^{3} + 6 x^{2} + 6 x - 2)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{10 h^{3} + 30 h^{2} x + 6 h^{2} + 30 h x^{2} + 12 h x + 6 h}{h}=10 h^{2} + 30 h x + 6 h + 30 x^{2} + 12 x + 6$