Answer
$2\left( x+2y \right)\left( 3x-2y \right)$.
Work Step by Step
Consider the expression.
$6{{x}^{2}}+8xy-8{{y}^{2}}$
Take out the greatest common factor, 2.
$6{{x}^{2}}+8xy-8{{y}^{2}}=2\left( 3{{x}^{2}}+4xy-4{{y}^{2}} \right)$
Factor the expression, $3{{x}^{2}}+4xy-4{{y}^{2}}$.
$\begin{align}
& 6{{x}^{2}}+8xy-8{{y}^{2}}=2\left( 3{{x}^{2}}+6xy-2xy-4{{y}^{2}} \right) \\
& =2\left( 3x\left( x+2y \right)-2y\left( x+2y \right) \right) \\
& =2\left( x+2y \right)\left( 3x-2y \right)
\end{align}$
Check:
$\begin{align}
& 2\left( x+2y \right)\left( 3x-2y \right)=2\left( x\cdot 3x-x\cdot 2y+2y\cdot 3x-2y\cdot 2y \right) \\
& =2\left( 3{{x}^{2}}-2xy+6xy-4{{y}^{2}} \right) \\
& =2\left( 3{{x}^{2}}+4xy-4{{y}^{2}} \right) \\
& =6{{x}^{2}}+8xy-8{{y}^{2}}
\end{align}$
