\( \operatorname { SimPlify } _ { ( x - 2 y ) ( x ^ { 2 } - 6 x y - y ^ { 2 } ) } \)

To simplify the expression \( (x - 2y)(x^2 - 6xy - y^2) \), start by distributing \( (x - 2y) \) across each term in the polynomial \( (x^2 - 6xy - y^2) \): 1. Multiply \( (x - 2y) \) by \( x^2 \): \[ x \cdot x^2 - 2y \cdot x^2 = x^3 - 2yx^2 \] 2. Multiply \( (x - 2y) \) by \( -6xy \): \[ x \cdot (-6xy) - 2y \cdot (-6xy) = -6x^2y + 12y^2x \] 3. Multiply \( (x - 2y) \) by \( -y^2 \): \[ x \cdot (-y^2) - 2y \cdot (-y^2) = -xy^2 + 2y^3 \] Now, combine all these terms: \[ x^3 - 2yx^2 - 6x^2y + 12y^2x - xy^2 + 2y^3 \] Next, group like terms: \[ x^3 + (-2y - 6y)x^2 + 12y^2x - xy^2 + 2y^3 \] \[ = x^3 - 8yx^2 + 12y^2x - xy^2 + 2y^3 \] So, the simplified expression is: \[ x^3 - 8yx^2 + 12y^2x - xy^2 + 2y^3 \]