How to present a vertical multiplication/addition

the xlop package does this sort of thing. It does warn that it uses "french conventions", but at least for multiplication it looks fine, to me.

disclaimer: i last did multiplication sums in school in the 1950s...

\documentclass{article}
\usepackage{xlop}
\begin{document}
\opmul{384}{56}\qquad
\end{document}

How about a simple tabular environment:

\documentclass{article}
\begin{document}
\begin{tabular}{cccc}
  & 1 & 2 & 3 \\
+ &   & 3 & 4 \\
\hline
  & 1 & 5 & 7 \\
\end{tabular}
\end{document}

If you need to change spacing, you can use the @ specifier which automatically puts arguments in braces as a space between columns (in this case a thin space \,).

\documentclass{article}
\begin{document}
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
  & 1 & 2 & 3 \\
+ &   & 3 & 4 \\
\hline
  & 1 & 5 & 7 \\
\end{tabular}
\end{document}

Some other spaces are a thick space \; and a medium space \:. It is also possible to avoid the onerous typing of repeating column types by using the *{}{}:

\begin{tabular}{c*{3}{@{\,}c}}

That produces c, and then 3 times @{\,}c, which combines to c@{\,}c@{\,}c@{\,}c.

I think the simplest (least complicated) solution is to use \phantom, as follows:

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}\begin{array}{c}
\phantom{\times99}384\\
\underline{\times\phantom{999}56}\\
\phantom{\times9}2304\\
\underline{\phantom\times1920\phantom9}\\
\phantom\times21504
\end{array}\end{equation*}
\end{document}

Here, amsmath is needed for the equation* environment only; if you used equation you'd need no \usepackage at all. array allows multiple-line equations; \\ starts a new line. And \phantom{foo} prints a space the width of the characters foo. I've usually found 9 to be a good stand-in for any digit, but (e.g. if you have a few 1s) you may need to fiddle with it a bit.