11.) \( 2 x^{2}-6 x y-8 y^{2} \)

To factor the expression \( 2x^2 - 6xy - 8y^2 \), you can start by factoring out the greatest common factor, which is 2: \( 2(x^2 - 3xy - 4y^2) \). Now, focus on factoring the quadratic \( x^2 - 3xy - 4y^2 \). Look for two numbers that multiply to \( -4y^2 \) and add to \( -3y \): these numbers are \( -4y \) and \( y \). Thus, it can be factored as: \( 2(x - 4y)(x + y) \). So the fully factored expression is: \( 2(x - 4y)(x + y) \). To double-check your factoring, you can use the FOIL method (First, Outside, Inside, Last) to expand it back. When you multiply \( 2(x - 4y)(x + y) \), you should get the original quadratic \( 2x^2 - 6xy - 8y^2 \) once again, confirming that your factorization is correct!