“多项式的长（或垂直）乘法（长乘法规则）”

Question

（1）如何将两个多项式垂直相乘？

\documentclass[11pt,a4paper]{article}
\usepackage{amsmath}
\usepackage{array}
\usepackage{booktabs}
\usepackage{polynom}
\newcommand*{\Ph}{\hphantom{)}}%

\begin{document}
 Example-(01) The distribution of the following product is arranged.
 \begin{flalign*}
 (x^{3}+2x^{2}+2x+1)(x^{2}-x+1)
 \end{flalign*}
 $\begin{array}{r@{} r@{} r r}
 x^3&{}+2x^2&{}+2x+ 4 \\
 \times(x^2&{}-x+1) &\\
 \cmidrule{1-3}
 \end{array}$
\end{document}