a) \(m{x^2} + m{y^2} - n{x^2} - n{y^2} \)
\(= m\left( {{x^2} + {y^2}} \right) - n\left( {{x^2} + {y^2}} \right) \)
\(= \left( {{x^2} + {y^2}} \right)\left( {m - n} \right).\)
b) \(40bc + 9cx - 24bx - 15{c^2}\)
\(= \left( {40bc - 15{c^2}} \right) + \left( {9cx - 24bx} \right)\)
\( = 5c\left( {8b - 3c} \right) + 3x\left( {3c - 8b} \right) \)
\( = \left( {8b - 3c} \right)\left( {5c - 3x.} \right)\)
c) \(a\left( {{b^2} + {c^2} - {a^2}} \right) + b\left( {{c^2} + {a^2} - {b^2}} \right) \)
\(= a{b^2} + a{c^2} - {a^3} + b{c^2} + b{a^2} - {b^3}\)
\( = \left( {a{b^2} + b{a^2}} \right) + \left( {a{c^2} + b{c^2}} \right) - \left( {{a^3} + {b^3}} \right)\)
\( = ab\left( {a + b} \right) + {c^2}\left( {a + b} \right) - \left( {a + b} \right)\left( {{a^2} - ab + {b^2}} \right)\)
\( = \left( {a + b} \right)\left( {ab + {c^2} - {a^2} + ab - {b^2}} \right) \)
\(= \left( {a + b} \right)\left[ {{c^2} + \left( { - {a^2} + 2ab - {b^2}} \right)} \right]\)
\( = \left( {a + b} \right)\left[ {{c^2} - {{\left( {a - b} \right)}^2}} \right] \)
\(= \left( {a + b} \right)\left( {c + a - b} \right)\left( {c - a + b} \right).\)
