Multiply: \( (x-2)(2 x-4)(x+6) \)

The expression \( (x-2)(2x-4)(x+6) \) can be simplified step-by-step. First, notice that \( 2x-4 \) can be factored as \( 2(x-2) \). This gives us: \[ (x-2)(2(x-2))(x+6) \] Then we can combine the first two factors: \[ 2(x-2)^2(x+6) \] Now, we can expand \( (x-2)^2 \): \[ (x-2)^2 = x^2 - 4x + 4 \] Now substitute this back into our expression: \[ 2(x^2 - 4x + 4)(x + 6) \] Next, we will expand this using the distributive property: 1. First multiply \( x^2 - 4x + 4 \) by \( x \): \[ x(x^2 - 4x + 4) = x^3 - 4x^2 + 4x \] 2. Then multiply \( x^2 - 4x + 4 \) by \( 6 \): \[ 6(x^2 - 4x + 4) = 6x^2 - 24x + 24 \] Now combine the two results: \[ x^3 - 4x^2 + 4x + 6x^2 - 24x + 24 = x^3 + ( -4x^2 + 6x^2) + (4x - 24x) + 24 \] \[ = x^3 + 2x^2 - 20x + 24 \] Finally, multiply everything by \( 2 \): \[ 2(x^3 + 2x^2 - 20x + 24) = 2x^3 + 4x^2 - 40x + 48 \] So the final simplified expression is: \[ \boxed{2x^3 + 4x^2 - 40x + 48} \]