How can I place text under a bracket\brace that isn't all one line

You could place the text in the second argument of \underbrace in a \parbox whose width is set to the width of the first argument of \underbrace. However, doing this manually can be tedious. Especially if you have several expressions like the one shown in your posting, it's a good idea to set up a dedicated macro to take care of the formatting chores. In the following example, this macro is called \bigbrace.

\documentclass{article}
\usepackage{amsmath,ragged2e}

\newlength\mylen
\newcommand\bigbrace[2]{%
   \settowidth\mylen{$\displaystyle #1$}
   \underbrace{#1}_{\parbox{\mylen}{\scriptsize\Centering #2}}}

\begin{document}
\[
\bigbrace{203948x^{3}+209204024x^2-18909345x}{How can I place a text
     that isn't on one long line, but aligned underneath the brace?}
\]
\end{document}

With stackengine for manual line breaks, or defining a \boxedunderbracecommand, based on the eqparbox package for an automatic line break: the text is placed in a \parbox of width equal to the formula width. It requires a tag as an optional argument (M by default) so as to differentiate multiple uses of \boxedunderbraces. Two compilations are necessary.

\documentclass{article}
\usepackage{amsmath,ragged2e}
\usepackage[usestackEOL]{stackengine}
\usepackage{eqparbox} 
\newcommand\boxedunderbrace[3] [M] {\underbrace{\eqmakebox[#1]{$\displaystyle#2$} }_{\scriptsize\parbox{\eqboxwidth{#1}}{\centering#3}}}

\begin{document}

\[
\underbrace{203948x^{3}+209204024x^2-18909345x}_{\scriptsize\Centerstack{How can
     I place a text that isn't on one long\\ line, but aligned underneath
     the brace?}}
\]

\[
\boxedunderbrace[M]{203948x^{3}+209204024x^2-18909345x}{How can
I place a text that isn't on one long line, but aligned underneath the brace?}
\]

\end{document}