\(a)\)\(( 5x - 2y )( x^2 - xy + 1)\)
\( = 5x.x^2 + 5x.(- xy) + 5x.1 \)\(+ (- 2y).x^2 + (- 2y).(- xy)+(-2y).1\)
\( = 5x^3 - 5x^2y + 5x - 2x^2y\) \(+ 2xy^2 - 2y\)
\( = 5{x^3} - 7{x^2}y + 5x + 2x{y^2} - 2y\)
\(b)\) \(\left( {x - 1} \right)\left( {x + 1} \right)\left( {x + 2} \right)\)
\( = \left( {x.x + x.1 + \left( { - 1} \right).x + \left( { - 1} \right).1} \right)\)\(.\left( {x + 2} \right)\)
\( = \left( {{x^2} + x - x - 1} \right)\left( {x + 2} \right)\)
\(= \left( {{x^2} - 1} \right)\left( {x + 2} \right)\)
\( = {x^2}.x + {x^2}.2 + \left( { - 1} \right).x + \left( { - 1} \right).2\)
\( = {x^3} + 2{x^2} - x - 2\)
\(c)\) \(\dfrac{1}{2}{x^2}{y^2}\left( {2x + y} \right)\left( {2x - y} \right)\)
\(= \dfrac{1}{2}x^2y^2(2x.2x + 2x.(- y) + y.2x \)\(+ y.(- y ) ) \)
\(=\dfrac{1}{2}{x^2}{y^2}\left( {4{x^2} - 2xy + 2xy - {y^2}} \right) \)
\(=\dfrac{1}{2}{x^2}{y^2}\left( {4{x^2} - {y^2}} \right) \)
\(\displaystyle = \dfrac{1}{2}{x^2}{y^2}.4{x^2} + {1 \over 2}{x^2}{y^2}.\left( { - {y^2}} \right) \)
\(\displaystyle = 2{x^4}{y^2} - {1 \over 2}{x^2}{y^4} \)
