\(\begin{array}{l}
a)\,\,\,1 + 2xy - {x^2} - {y^2} = 1 - \left( {{x^2} - 2xy + {y^2}} \right)\\
= 1 - {\left( {x - y} \right)^2} = \left( {1 - x + y} \right)\left( {1 + x - y} \right).\\
b)\,\,{a^2} + {b^2} - {c^2} - {d^2} - 2ab + 2cd\\
= \left( {{a^2} - 2ab + {b^2}} \right) - \left( {{c^2} - 2cd + {d^2}} \right)\\
= {\left( {a - b} \right)^2} - {\left( {c - d} \right)^2}\\
= \left( {a - b - c + d} \right)\left( {a - b + c - d} \right).\\
c)\,\,{a^3}{b^3} - 1 = \left( {ab - 1} \right)\left[ {{{\left( {ab} \right)}^2} - ab + 1} \right].\\
d)\,\,{x^2}\left( {y - z} \right) + {y^2}\left( {z - x} \right) + {z^2}\left( {x - y} \right)\\
= {x^2}\left( {y - z} \right) + {y^2}\left( {z - y + y - x} \right) + {z^2}\left( {x - y} \right)\\
= {x^2}\left( {y - z} \right) + {y^2}\left( {z - y} \right) + {y^2}\left( {y - x} \right) + {z^2}\left( {x - y} \right)\\
= {x^2}\left( {y - z} \right) - {y^2}\left( {y - z} \right) + {y^2}\left( {y - x} \right) - {z^2}\left( {y - x} \right)\\
= \left( {y - z} \right)\left( {{x^2} - {y^2}} \right) - \left( {x - y} \right)\left( {{y^2} - {z^2}} \right)\\
= \left( {y - z} \right)\left( {x - y} \right)\left( {x + y} \right) - \left( {x - y} \right)\left( {y - z} \right)\left( {y + z} \right)\\
= \left( {y - z} \right)\left( {x - y} \right)\left( {x + y - y - z} \right)\\
= \left( {y - z} \right)\left( {x - y} \right)\left( {x - z} \right).\\
e)\,\,{x^2} - 15x + 36\\
= {x^2} - 3x - 12x + 36\\
= x\left( {x - 3} \right) - 12\left( {x - 3} \right)\\
= \left( {x - 3} \right)\left( {x - 12} \right).\\
f)\,\,\,{x^{12}} - 3{x^6}{y^6} + 2{y^{12}}\\
= {x^{12}} - {x^6}{y^6} - 2{x^6}{y^6} + 2{y^{12}}\\
= {x^6}\left( {{x^6} - {y^6}} \right) - 2{y^6}\left( {{x^6} - {y^6}} \right)\\
= \left( {{x^6} - {y^6}} \right)\left( {{x^6} - 2{y^6}} \right)\\
= \left( {{x^3} + {y^3}} \right)\left( {{x^3} - {y^3}} \right)\left( {{x^6} - 2{y^6}} \right)\\
= \left( {x + y} \right)\left( {{x^2} - xy + {y^2}} \right)\left( {x - y} \right)\left( {{x^2} + xy + {y^2}} \right)\left( {{x^6} - 2{y^6}} \right).\\
g)\,\,{x^8} - 64{x^2} = {x^2}\left( {{x^4} - 64} \right)\\
= {x^2}\left( {{x^2} - 8} \right)\left( {{x^2} + 8} \right).\\
h)\,\,\,{\left( {{x^2} - 8} \right)^2} - 784 = \left( {{x^2} - 8 - 28} \right)\left( {{x^2} - 8 + 28} \right)\\
= \left( {{x^2} - 36} \right)\,\left( {{x^2} + 20} \right) = \left( {x - 6} \right)\left( {x + 6} \right)\left( {{x^2} + 20} \right).
\end{array}\)
