Here are a couple of suggestions:
When using pdfpages each included page will be scaled from the source document - completenotes310pages.pdf in your case - to the destination document <jobname>.pdf unless you specify the noautoscale key. Even if you do not specify this, or if the pages you're importing are of the exact size that you're currently working on, once a page is full, LaTeX will ship it out, and move on to the following page. This is the case with the MWE, even though you had hoped hello would be printed on page 1. This leads to the following suggestion...
You will have to include the pages on an as-needed basis in order to "pause" LaTeX momentarily and add content to the page. For example, you may have a document sequence that looks like this:
\documentclass{article}
\usepackage{pdfpages}% http://ctan.org/pkg/pdfpages
\begin{document}
\includepdf[pages=1-5]{completenotes310pages.pdf}% Includes pages 1-5 (no additions on these pages)
<some stuff>
\includepdf[pages=6]{completenotes310pages.pdf}% Includes page 6
<some more stuff>
\includepdf[pages=7-310]{completenotes310pages.pdf}% Includes pages 7-310 (no additions on these pages)
\end{document}
where <some stuff> and <some more stuff> detail your mathematical addition.
If the original document is NOT typeset in LaTeX, and it has whitespace that is actually a picture, you may have to typeset the page in the background, before overlaying it with LaTeX mathematics. For this, I would suggest using the everyshi package that provides \AtNextShipout{...} or the eso-pic package command \AddToShipoutBG{...}. Actually, I think the latter option would work better in this case. And you could switch to using the foreground or background options ...FG or ...BG depending on whether you're placing the math content or the page.
Here is a practical example illustrating some of the suggestions above:
Consider the following source document, called source.tex and output source.pdf:
\documentclass{article}
\usepackage{lipsum}
\begin{document}
\lipsum[1-2]
\vspace{5cm}
\lipsum[3-4]
\end{document}

You'll notice the gap that requires some filling. This is how I would do it:
\documentclass{article}
\usepackage{pdfpages}% http://ctan.org/pkg/pdfpages
\usepackage{eso-pic}% http://ctan.org/pkg/eso-pic
\pagestyle{empty}
\begin{document}
% Add pages before
\AddToShipoutPictureFG*{%
\AtPageCenter{%
\vspace{2cm}
\makebox[0pt][c]{\Huge HERE IS SOME TEXT.}
}
}%
\includepdf[pages=1]{source.pdf}
% Add pages after
\end{document}

completenotes310pages.pdfhas blank spaces where you want to put the math content? And you want to put the math content in the specific spaces using LaTeX? – Werner Sep 03 '11 at 19:58