Consider
a/b = c/d = e/f = k
So we get
a = bk, c = dk, e = fk
Therefore, LHS = RHS.
So we get
\[=\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1\text{ }+\text{ }k\text{ }+\text{ }1\text{ }+\text{ }k\text{ }+\text{ }1 \right)}^{3}}\]
By further calculation
\[\begin{array}{*{35}{l}}
=\text{ }bdf\text{ }{{\left( 3k\text{ }+\text{ }3 \right)}^{3}} \\
=\text{ }27\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1 \right)}^{3}} \\
\end{array}\]
RHS = \[27\text{ }\left( a\text{ }+\text{ }b \right)\text{ }\left( c\text{ }+\text{ }d \right)\text{ }\left( e\text{ }+\text{ }f \right)\]
It can be written as
\[=\text{ }27\text{ }\left( bk\text{ }+\text{ }b \right)\text{ }\left( dk\text{ }+\text{ }d \right)\text{ }\left( fk\text{ }+\text{ }f \right)\]
Taking out the common terms
\[=\text{ }27\text{ }b\text{ }\left( k\text{ }+\text{ }1 \right)\text{ }d\text{ }\left( k\text{ }+\text{ }1 \right)\text{ }f\text{ }\left( k\text{ }+\text{ }1 \right)\]
So we get
\[=\text{ }27\text{ }bdf\text{ }{{\left( k\text{ }+\text{ }1 \right)}^{3}}\]
Therefore, LHS = RHS.