[Tex/LaTex] Multi-line Equations

Question

I am trying to split my equation. I am using the code below. However, each time I am getting the following error

! Extra }, or forgotten \right. } $\hfill \endtemplate
l.215 \end{split} ?

\documentclass{article}
\usepackage{amsmath}    

\begin{document}
\begin{equation}\begin{split}
MSE_{II}(\overline{y}_{CST})&= MSE_{I}(\overline{y}_{CST})+\overline{Y}^{2}\left[\left(c_{2}^{2}+2c_{1}c_{3}\right)V_{4,\, 0}+2c_{1}c_{2}V_{3,\, 0}+2\left(2c_{1}c_{2}+c_{3}\right)V_{3,\, 1} +\\ 
&+2 \left(c_{1}^{2}+c_{2} \right)V_{2,\, 1}+\left( c_{1}^{2}+2c_{2} \right)V_{2,\, 2}+{2c_{1}V_{1,\, 2}}\right]
\end{split}\end{equation}
\end{document}

But, when I remove, \left[ and \right] codes, it works fine. I do not understand. Please, help me.

Answer 1

Besides the balancing issue associated to the use of \left, \right, your expression can be improved introducing some modifications:

1. Using \left( ... \right) for the inner expressions produces delimiters that are too big and introduce some additional horizontal space that it's superfluous in this particular case, so I think that it's better to use just (...).
2. You need to use +{} so that the ending plus sign in the first line is correctly trated as an operator; analogously, you need to use {}+ for the first plus sign in the second line.
3. Instead of using \left[, \right. in the first line and \right., \left] in the second line, I would suggest you to use \bigl[, \bigr] which don't need to be balanced in each line and produce better sized delimiters in most cases.
4. The plus sign in the second line must be (according to some standard typographical rules for displayed math expressions) two-ems to the right of the alignment point.

In the following code I include, for comparison, your code (with the balancing problem solved) and then the code implementing my suggestions:

\documentclass[12pt]{article}
\usepackage[margin=2cm]{geometry}% just for the example
\usepackage{amsmath}

\begin{document}

\begin{equation}
\begin{split}
MSE_{II}(\overline{y}_{CST}) &= MSE_{I}(\overline{y}_{CST})
  +\overline{Y}^{2}\left[ \left( c_{2}^{2}+2c_{1}c_{3} \right)V_{4,\, 0}+2c_{1}c_{2}V_{3,\,0}
  +2\left(2c_{1}c_{2}+c_{3}\right)V_{3,\, 1} + \right. \\ 
& \left. + 2 \left(c_{1}^{2}+c_{2} \right)V_{2,\, 1}+ \left(c_{1}^{2}+2c_{2} \right)V_{2,\, 2} 
 + {2c_{1}V_{1,\, 2}}  \right]
\end{split}
\end{equation}

\begin{equation}
\begin{split}
MSE_{II} ( \overline{y}_{CST} ) &= MSE_{I} ( \overline{y}_{CST} )
  + \overline{Y}^{2} \bigl[ (c_{2}^{2} + 2c_{1}c_{3} ) V_{4,\, 0} + 2c_{1}c_{2} V_{3,\,0}
  + 2 ( 2c_{1}c_{2} + c_{3} ) V_{3,\, 1} +{} \\ 
&\qquad {}+ 2 ( c_{1}^{2} + c_{2} )V_{2,\, 1} + ( c_{1}^{2} + 2c_{2} ) V_{2,\, 2} 
 + {2c_{1}V_{1,\, 2}} \bigr]
\end{split}
\end{equation}

\end{document}