Example 1 Given $x+y=3, x^{2}+y^{2}-x y=4$. Then the value of $x^{4}+y^{4}+x^{3} y+x y^{3}$ is $\qquad$
(13th Junior High School Mathematics Competition of Jiangsu Province)
$$
\begin{array}{l}
\text { Solution: } x^{4}+y^{4}+x^{3} y+x y^{3} \\
=(x+y)^{2}\left(x^{2}+y^{2}-x y\right)
\end{array}
$$

Substitute $x+y=3, x^{2}+y^{2}-x y=4$, we get
$$
x^{4}+y^{4}+x^{3} y+x y^{3}=3^{2} \times 4=36 .
$$