a) \({x^3} + {y^3} + x + y = \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right) + \left( {x + y} \right) = \left( {x + y} \right)\left( {{x^2} - xy + {y^2} + 1} \right)\)
b) \({x^3} - {y^3} + x - y = \left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right) + \left( {x - y} \right) = \left( {x - y} \right)\left( {{x^2} + xy + {y^2} + 1} \right)\)
c)
\(\begin{array}{l}{\left( {x - y} \right)^3} + {\left( {x + y} \right)^3} = \left( {x - y + x + y} \right)\left[ {{{\left( {x - y} \right)}^2} - \left( {x - y} \right)\left( {x + y} \right) + {{\left( {x + y} \right)}^2}} \right]\\ = 2x.\left( {{x^2} - 2xy + {y^2} - {x^2} + {y^2} + {x^2} + 2xy + {y^2}} \right) = 2x\left( {{x^2} + 3{y^2}} \right)\end{array}\)
d)
\(\begin{array}{l}{x^3} - 3{x^2}y + 3x{y^2} - {y^3} + {y^2} - {x^2} = \left( {{x^3} - 3{x^2}y + 3x{y^2} - {y^3}} \right) + \left( {{y^2} - {x^2}} \right)\\ = {\left( {x - y} \right)^3} + \left( {y - x} \right)\left( {y + x} \right) = \left( {x - y} \right)\left[ {{{\left( {x - y} \right)}^2} - y - x} \right] = \left( {x - y} \right)\left( {{x^2} - 2xy + {y^2} - x - y} \right)\end{array}\)