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I want to write an equation and put some additional information in the equation but left of the equation number, thus \tag{} does not work. Furthermore, I would like to have the description flushed right so that it looks neat with multiple equations beneath.

For example, I would like something like for all i standing left of my equation number, flushed to the right side (in the red box in the picture):

enter image description here

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You can use flalign with two "equations" in order to write a comment. Here's an example:

enter image description here

\documentclass{article}

\usepackage[fleqn]{amsmath}
\setlength{\mathindent}{0pt}% No indent for full length equations (flush left)

\begin{document}

\noindent X \dotfill X
\begin{flalign}
  f(x) &= ax^2 + bx + c && \text{Comment} \label{eqn:comment}
\end{flalign}

\noindent X \dotfill X
\begin{align}
  f(x) &= ax^2 + bx + c \label{eqn:nocomment}
\end{align}

See~\eqref{eqn:comment} and~\eqref{eqn:nocomment}.

\end{document}

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