%% 
%%  A DANTE-Edition example
%% 
%%  Example 36-00-17 on page 775.
%% 
%%  Copyright (C) 2011 Herbert Voss
%% 
%%  It may be distributed and/or modified under the conditions
%%  of the LaTeX Project Public License, either version 1.3
%%  of this license or (at your option) any later version.
%% 
%%  See http://www.latex-project.org/lppl.txt for details.
%% 
%% 
%% ==== 
% Show page(s) 1
%% 
\documentclass[]{article}
\pagestyle{empty}
\setlength\textwidth{201.70511pt}
\setlength\parindent{0pt}

\usepackage{pst-node,multido,amsmath}\newdimen\xMax
\newdimen\yMax
\newcount\xLines
\newcount\yLines
\newdimen\dx
\newdimen\dy
\definecolor{lightred}{rgb}{1.0, 0.8, 0.8}
\makeatletter
\newcommand\grid[3][black]{{%
  \psset{linewidth=0.1pt}
  \xMax=#3%
  \yMax=#2%
    \dx=5mm  \xLines=\xMax \divide\xLines by \dx%
    \dy=5mm  \yLines=\yMax \divide\yLines by \dy%
    \advance\xLines by 1%
    \advance\yLines by 1%
    \yMax=\dy \multiply\yMax by \yLines%
    \xMax=\dx \multiply\xMax by \xLines%
    \advance\xLines by 1%
    \advance\yLines by 1%
    \psset{unit=1pt, linecolor=#1}%
    \multido{\rA=0+\strip@pt\dx}{\xLines}{\psline(\rA,0)(\rA,\yMax)}%
    \multido{\rA=0+\strip@pt\dy}{\yLines}{\psline(0,\rA)(\xMax,\rA)}%
}}
\makeatother
\newsavebox{\gridbox}
\newenvironment{dogrid}[1][\linewidth]{%
  \begin{lrbox}{\gridbox}%
      \begin{minipage}{#1}%
}{%
      \end{minipage}%
  \end{lrbox}%
  \yMax=\dp\gridbox \advance\yMax by \ht\gridbox
  \noindent%
  \raisebox{-1.05\dp\gridbox}{\grid[lightred]{\yMax}{\wd\gridbox}}%
  \usebox{\gridbox}
  \vspace{0.5cm}
}


\begin{document}
\begin{dogrid}
\[
\begin{array}{rcll}
y & = & x^{2}+bx+c\\
  & = & x^{2}+2\cdot{\displaystyle\frac{b}{2}x+c}\\
  & = & \underbrace{x^{2}+2\cdot\frac{b}{2}x+
            \left(\frac{b}{2}\right)^{2}}-{\displaystyle \left(\frac{b}{2}\right)^{2}+c}\\
 &  & \qquad\color{blue}\left(x+{\displaystyle \frac{b}{2}}\right)^{2}\\
 & = & {\color{blue}\left(x+{\displaystyle \frac{b}{2}}\right)^{2}}
   \color{red}-\left({\displaystyle \frac{b}{2}}\right)^{2}+c & \left|+\left({\displaystyle \frac{b}{2}}\right)^{2}-c\right.\\
y+\left({\displaystyle \frac{b}{2}}\right)^{2}-c & = & \left(x+{\displaystyle \frac{b}{2}}\right)^{2} & \left|(\textrm{Scheitelpunktform})\right.\\
y-y_{S} & = & (x-x_{S})^{2}\\
\mbox{\textbf{S}}\mathbf{\left(x_{S};y_{S}\right)} & \,\textrm{bzw.}\, &
    \mbox{\textbf{S}}\mathbf{\left(-{\displaystyle \frac{b}{2};\,
    \left({\displaystyle \frac{b}{2}}\right)^{2}-c}\right)}
\end{array}
\]
\end{dogrid}
\end{document}