Answer
$4x^{3}+6x^{2}h+4xh^{2}+h^{3}$
Work Step by Step
We apply the binomial theorem to expand $(x+h)^3$:
$(x+h)^4=\left(\begin{array}{l}4\\0\end{array}\right)x^{4}+\left(\begin{array}{l}
4\\1\end{array}\right)x^{3}h^{1}+\left(\begin{array}{l}
4\\2\end{array}\right)x^{2}h^{2}+\left(\begin{array}{l}
4\\3\end{array}\right)x^{1}h^{3}+\left(\begin{array}{l}
4\\4\end{array}\right)h^{4}=1x^{4}+4x^{3}h^1+6x^{2}h^{2}+4x^1h^{3}+1h^{4}$
Now we simplify:
$\displaystyle \frac{(x+h)^{4}-x^{4}}{h}=\frac{x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}-x^{4}}{h}
=\frac{4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}}{h}=\frac{h(4x^{3}+6x^{2}h+4xh^{2}+h^{3})}{h}=4x^{3}+6x^{2}h+4xh^{2}+h^{3}$
