Multiple tags/right-justified text in unaligned environments using amsmath

How can I use multiple tags or right-justified text elements in an equation environment with many lines while keeping all equations centered? Unfortunately, I can only use packages accepted by the ACM TAPS system. It would be great if you could provide answers for the aforementioned package list. But also general answers are very welcome!

So far I have the following:

begin{gather*}
begin{split}
 d(x) + e(x) + f(x) < a(x) * b(x) * c(x)  
      * d(x) * e(x) * f(x)  hfilltext{Explained here}
end{split}
begin{split}
b(x) + c(x) + d(x) + e(x) + f(x) < a(x) * b(x) * c(x)  
      * d(x) * e(x) * f(x)  hfilltext{And explained further here}
end{split}
end{gather*}

This keeps the equations centered, however, the text is not right justified but aligned to the right of the equation:

Also if I use the align environment, the equations are right-justified even without using an &:

begin{align*}
d(x) + e(x) + f(x) < a(x) * b(x) * c(x) * d(x) 
      * e(x) * f(x) tag*{Explained here} 
a(x) + b(x) + c(x) + d(x) + e(x) + f(x) < a(x) * b(x) * c(x)  
      * d(x) * e(x) * f(x) tag*{And explained further here}
end{align*}

Note: I don’t want to use the & because the equations have different lengths on the left side.

documentclass{article}
usepackage{mathtools}

begin{document}

begin{gather*}
begin{split}
 d(x) + e(x) + f(x) < a(x) * b(x) * c(x) 
     {} * d(x) * e(x) * f(x)
end{split}
shortintertext{raggedleft Explained here}
begin{split}
b(x) + c(x) + d(x) + e(x) + f(x) < a(x) * b(x) * c(x) 
     {} * d(x) * e(x) * f(x) 
end{split}
shortintertext{raggedleft And explained further here}
end{gather*}

end{document} 

   begin{gather*}
    begin{split}
     d(x) + e(x) + f(x) < a(x) * b(x) * c(x) 
         {} * d(x) * e(x) * f(x)
    end{split}
    makebox[textwidth][r]{Explained here}
    begin{split}
    b(x) + c(x) + d(x) + e(x) + f(x) < a(x) * b(x) * c(x) 
         {} * d(x) * e(x) * f(x)
    end{split}
    makebox[textwidth][r]{And explained further here}
    end{gather*}