\(a)\) \({x^4} + 2{x^3} + {x^2}\)
\( = {x^2}\left( {{x^2} + 2x + 1} \right)\)
\( = {x^2}{\left( {x + 1} \right)^2}\)
\(b)\) \({x^3} - x + 3{x^2}y + 3x{y^2} + {y^3} – y\)
\(= \left( {{x^3} + 3{x^2}y + 3x{y^2} + {y^3}} \right) - \left( {x + y} \right) \)
\(= {\left( {x + y} \right)^3} - \left( {x + y} \right)\)
\(= \left( {x + y} \right)\left[ {{{\left( {x + y} \right)}^2} - 1} \right]\)
\( = \left( {x + y} \right)\left( {x + y + 1} \right)\left( {x + y - 1} \right) \)
\(c)\) \(5{x^2} - 10xy + 5{y^2} - 20{z^2} \)
\(= 5\left( {{x^2} - 2xy + {y^2} - 4{z^2}} \right)\)
\( = 5\left[ {\left( {{x^2} - 2xy + {y^2}} \right) - 4{z^2}} \right] \)
\(= 5\left[ {{{\left( {x - y} \right)}^2} - {{\left( {2z} \right)}^2}} \right] \)
\(= 5\left( {x - y + 2z} \right)\left( {x - y - 2z} \right) \)
