Giải thích các bước giải:

Ta có:

\(\begin{array}{l}
a,\\
16 - {x^2} + 2xy - {y^2}\\
 = 16 - \left( {{x^2} - 2xy + {y^2}} \right)\\
 = {4^2} - {\left( {x - y} \right)^2}\\
 = \left( {4 - x + y} \right)\left( {4 + x - y} \right)\\
b,\\
{x^2} - xy - 3x + 3y\\
 = \left( {{x^2} - xy} \right) - \left( {3x - 3y} \right)\\
 = x\left( {x - y} \right) - 3\left( {x - y} \right)\\
 = \left( {x - y} \right)\left( {x - 3} \right)\\
c,\\
{x^2} + 2x - 4{y^2} - 4y\\
 = \left( {{x^2} + 2x + 1} \right) - \left( {4{y^2} + 4y + 1} \right)\\
 = {\left( {x + 1} \right)^2} - {\left( {2y + 1} \right)^2}\\
 = \left[ {\left( {x + 1} \right) - \left( {2y + 1} \right)} \right].\left[ {\left( {x + 1} \right) + \left( {2y + 1} \right)} \right]\\
 = \left( {x - 2y} \right)\left( {x + 2y + 2} \right)\\
d,\\
{x^4} - 6{x^3} + 54x - 81\\
 = \left( {{x^4} - 81} \right) - \left( {6{x^3} - 54x} \right)\\
 = \left( {{x^2} - 9} \right)\left( {{x^2} + 9} \right) - 6x\left( {{x^2} - 9} \right)\\
 = \left( {{x^2} - 9} \right).\left( {{x^2} + 9 - 6x} \right)\\
 = \left( {x - 3} \right)\left( {x + 3} \right){\left( {x - 3} \right)^2}\\
 = {\left( {x - 3} \right)^3}\left( {x + 3} \right)\\
e,\\
a{x^2} + ax - b{x^2} - bx - a + b\\
 = \left( {a{x^2} - b{x^2}} \right) + \left( {ax - bx} \right) - \left( {a - b} \right)\\
 = {x^2}\left( {a - b} \right) + x\left( {a - b} \right) - \left( {a - b} \right)\\
 = \left( {a - b} \right)\left( {{a^2} + x - 1} \right)\\
g,\\
{\left( {{x^2} + {y^2} - 2} \right)^2} - {\left( {2xy - 2} \right)^2}\\
 = \left[ {\left( {{x^2} + {y^2} - 2} \right) - \left( {2xy - 2} \right)} \right].\left[ {\left( {{x^2} + {y^2} - 2} \right) + \left( {2xy - 2} \right)} \right]\\
 = \left( {{x^2} - 2xy + {y^2}} \right).\left( {{x^2} + {y^2} + 2xy - 4} \right)\\
 = {\left( {x - y} \right)^2}.\left[ {{{\left( {x + y} \right)}^2} - 4} \right]\\
 = {\left( {x - y} \right)^2}.\left( {x + y - 2} \right)\left( {x + y + 2} \right)
\end{array}\)