Matematyka.pl

\begin{equation}
\begin{split}
a^{3}b+ a^{3}c +b ^{3}a+b ^{3}c+c ^{3}a+c ^3{b} - 2a ^{2}bc-2b ^{2}ac-2c ^{2} ab & = \\
& = (ab+c^2)(a-b)^2 + (bc+a^2)(b-c)^2 + (ac+b^2)(a-c)^2.
\end{split}
\end{equation}

\begin{equation}
\begin{split}
a^{3}b+ a^{3}c +b ^{3}a+b ^{3}c+c ^{3}a+c ^3{b} -\orange{ 2a ^{2}bc}-\orange{ 2b ^{2}ac}-\orange{ 2c ^{2} ab} & = \red{a^{3}b +b^3a}+\blue{ a^3c+c^3a}+\green{b^3c+c^3b } \\
& \quad -\red{2a^2b^2} - \blue{2a^2c^2} - \green{2b^2c^2} \\
& \quad + \red{a^2b^2} +\red{a^2c^2} + \blue{a^2b^2} + \blue{b^2c^2}+\green{a^2c^2}+\green{b^2c^2}\\
& \quad -\orange{ 2a ^{2}bc}-\orange{ 2b ^{2}ac}-\orange{ 2c ^{2} ab} \\
&= ab\left( a-b \right)^2 +ac\left( a-c\right)^2 + bc\left( b-c\right)^2 \\
&\quad + a^2(b-c)^2+b^2(a-c)^2+c^2(a-b)^2 \\
&= (ab+c^2)(a-b)^2 + (bc+a^2)(b-c)^2 + (ac+b^2)(a-c)^2.
\end{split}
\end{equation}