Answer
$4x^{3}+6x^{2}h+4xh^{2}+h^{3}$
Work Step by Step
In the numerator, expand
$(x+h)^{4}=\left(\begin{array}{l}
4\\
0
\end{array}\right)x^{4}h^{0}+\left(\begin{array}{l}
4\\
1
\end{array}\right)x^{3}h^{1}+\left(\begin{array}{l}
4\\
2
\end{array}\right)x^{3}h^{2}+\left(\begin{array}{l}
4\\
3
\end{array}\right)x^{1}h^{3}+\left(\begin{array}{l}
4\\
4
\end{array}\right)h^{4}$
$\displaystyle \frac{(x+h)^{4}-x^{4}}{h}=\frac{x^{4}+4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}-x^{4}}{h}$
$=\displaystyle \frac{4x^{3}h+6x^{2}h^{2}+4xh^{3}+h^{4}}{h}$
..factor out h in the numerator...
$=\displaystyle \frac{h(4x^{3}+6x^{2}h+4xh^{2}+h^{3})}{h}$
... reduce ...
$=4x^{3}+6x^{2}h+4xh^{2}+h^{3}$
