QUESTION 2 \( 2.1 \quad \) Smmplify the following expressions: \( \begin{array}{l}2.1 .1 \quad(x-2 y)\left(2 x^{2}+x y-y^{2}\right) \\ 2.1 .2 \frac{x^{2}-x+1}{x^{3}+1}+\frac{2}{x^{2}-1}\end{array} \)

To simplify the first expression \( (x-2y)(2x^2+xy-y^2) \), start by distributing \( (x-2y) \) to each term in the second polynomial. You'll have: \[ x(2x^2) + x(xy) - x(y^2) - 2y(2x^2) - 2y(xy) + 2y(y^2) \] This simplifies to: \[ 2x^3 + xy^2 - xy^2 - 4xy^2 + 2y^3 \] Combining like terms gives: \[ 2x^3 - 3xy^2 + 2y^3 \] For the second expression \( \frac{x^2-x+1}{x^3+1} + \frac{2}{x^2-1} \), start by factoring the denominators: \( x^3 + 1 = (x+1)(x^2-x+1) \) and \( x^2-1 = (x-1)(x+1) \). The least common denominator will be \( (x+1)(x^2-x+1)(x-1) \). Combining the fractions involves adjusting each fraction's numerator accordingly. After simplification, you'll get: \[ \frac{(x^2-x+1)(x-1) + 2(x^2-x+1)}{(x+1)(x^2-x+1)(x-1)} \] Simplify the numerator to arrive at the final expression.