Đáp án:

$\begin{array}{l}
a)\left( {x - y} \right)\left( {{x^3} + {x^2}y + x{y^2} + {y^3}} \right)\\
 = {x^4} + {x^3}y + {x^2}{y^2} + x{y^3} - {x^3}y - {x^2}{y^2} - x{y^3} - {y^4}\\
 = {x^4} - {y^4}\\
b)\left( {x + y} \right)\left( {{x^4} - {x^3}y + {x^2}{y^2} - x{y^3} + {y^4}} \right)\\
 = {x^5} - {x^4}y + {x^3}{y^2} - {x^2}{y^3} + x{y^4} + {x^4}y - {x^3}{y^2}\\
 + {x^2}{y^3} - x{y^4} + {y^5}\\
 = {x^5} + {y^5}\\
c)\left( {x - y} \right)\left( {{x^4} + {x^3}y + {x^2}{y^2} + x{y^3} + {y^4}} \right)\\
 = {x^5} + {x^4}y + {x^3}{y^2} + {x^2}{y^3} + x{y^4}\\
 - {x^4}y - {x^3}{y^2} - {x^2}{y^3} - x{y^4} - {y^5}\\
 = {x^5} - {y^5}\\
d)\left( {x - 1} \right)\left( {{x^2} + x + 1} \right)\\
 = {x^3} + {x^2} + x - {x^2} - x - 1\\
 = {x^3} - 1\\
e){x^2} + {y^2} + {z^2} + 2xy + 2yz + 2xz\\
 = {x^2} + 2xy + {y^2} + {z^2} + 2yz + 2xz\\
 = {\left( {x + y} \right)^2} + 2\left( {x + y} \right)z + {z^2}\\
 = {\left( {x + y + z} \right)^2}\\
f){\left( {x + y + z} \right)^3}\\
 = {\left( {\left( {x + y} \right) + z} \right)^3} = {\left( {x + y} \right)^3} + 3{\left( {x + y} \right)^2}z + 3\left( {x + y} \right){z^2} + {z^3}\\
 = {x^3} + 3{x^2}y + 3x{y^3} + {y^3} + 3{x^2}z + 6xyz + 3{y^2}z + 3x{z^2} + 3y{z^2} + {z^3}\\
 = {x^3} + {y^3} + {z^3} + 3\left( {{x^2}y + x{y^3} + {x^2}z + 2xyz + {y^2}z + x{z^2} + y{z^2}} \right)\\
 = {x^3} + {y^3} + {z^3} + 3\left( {x + y} \right)\left( {y + z} \right)\left( {x + z} \right)\\
g)\left( {x + y + z} \right)\left( {{x^2} + {y^2} + {z^2} - xy - yz - xz} \right)\\
 = {x^3} + {y^3} + {z^3} - {x^2}y - xyz - {x^2}z - x{y^2} - {y^2}z - xyz\\
 - xyz - y{z^2} - x{z^2}\\
 = {x^3} + {y^3} + {z^3} - 3xyz
\end{array}$