Find the difference quotient of $LaTeX: \displaystyle f(x)=4 x^{3} - 3 x^{2} + 5 x - 8$ .

The difference quotient is $LaTeX: \displaystyle \frac{f(x+h)-f(x)}{h}$ . Evaluating $LaTeX: \displaystyle f(x+h)=5 h + 5 x + 4 \left(h + x\right)^{3} - 3 \left(h + x\right)^{2} - 8$ and expanding gives $LaTeX: \displaystyle f(x+h)=4 h^{3} + 12 h^{2} x - 3 h^{2} + 12 h x^{2} - 6 h x + 5 h + 4 x^{3} - 3 x^{2} + 5 x - 8$ Evaluating the difference quotient gives $LaTeX: \displaystyle \frac{(4 h^{3} + 12 h^{2} x - 3 h^{2} + 12 h x^{2} - 6 h x + 5 h + 4 x^{3} - 3 x^{2} + 5 x - 8)-(4 x^{3} - 3 x^{2} + 5 x - 8)}{h}$ Simplifying gives $LaTeX: \displaystyle \frac{4 h^{3} + 12 h^{2} x - 3 h^{2} + 12 h x^{2} - 6 h x + 5 h}{h}=4 h^{2} + 12 h x - 3 h + 12 x^{2} - 6 x + 5$