Step-by-step explanation:
\usepackage{amsmath}
\begin{document}
one centered formula, without label:
\begin{equation*}
a x^2 + b x + c = 0
\end{equation*}
one centered formula, with label:
\begin{equation}
a x^2 + b x + c = 0
\end{equation}
several centered formulas, without label:
\begin{gather*}
a x + b = 0 \\
a x^2 + b x + c = 0 \\
a x^3 + b x^2 + c x + d = 0
\end{gather*}
several centered formulas, one label for all of them:
\begin{equation}
\begin{gathered}
a x + b = 0 \\
a x^2 + b x + c = 0 \\
a x^3 + b x^2 + c x + d = 0
\end{gathered}
\end{equation}
several centered formulas, each with its own label
\begin{gather}
a x + b = 0 \\
a x^2 + b x + c = 0 \\
a x^3 + b x^2 + c x + d = 0
\end{gather}
several formulas, any alignment, without label:
\begin{flalign*}
10xy^2+15x^2y-5xy & = 5\left(2xy^2+3x^2y-xy\right) = \\
& = 5x\left(2y^2+3xy-y\right) = \\
& = 5xy\left(2y+3x-1\right)
\end{flalign*}
several formulas, any alignment, each with its own label:
\begin{flalign}
10xy^2+15x^2y-5xy & = 5\left(2xy^2+3x^2y-xy\right) = \\
& = 5x\left(2y^2+3xy-y\right) = \\
& = 5xy\left(2y+3x-1\right)
\end{flalign}
several formulas, any alignment, one label for all of them
\begin{equation}
\begin{split}
10xy^2+15x^2y-5xy & = 5\left(2xy^2+3x^2y-xy\right) = \\
& = 5x\left(2y^2+3xy-y\right) = \\
& = 5xy\left(2y+3x-1\right)
\end{split}
\end{equation}
splitting a long formula on several lines. The first line is left-aligned, the last one is right-aligned, all the others are centered:.
\begin{multline}
\left(1+x\right)^n = 1 + nx + \frac{n\left(n-1\right)}{2!}x^2 +\\
+ \frac{n\left(n-1\right)\left(n-2\right)}{3!}x^3 +\\
+ \frac{n\left(n-1\right)\left(n-2\right)\left(n-3\right)}{4!}x^4 + \dots
\end{multline}
subordinate numbering:
\begin{subequations}
\begin{gather}
a x + b = 0 \\
a x^2 + b x + c = 0 \\
a x^3 + b x^2 + c x + d = 0
\end{gather}
\end{subequations}
boxed formula:
\begin{equation*}
\boxed{a x^2 + b x + c = 0}
\end{equation*}
\end{document}
