Thực hiện phép tính:

a) \(\left( {{x^4} + 2x{y^2} + \frac{1}{2}} \right) + \left( {2{x^4} - x{y^2} - 1} \right)\)

\[ = {x^4} + 2x{y^2} + \frac{1}{2} + 2{x^4} - x{y^2} - 1\]

\[ = \left( {{x^4} + 2{x^4}} \right) + \left( {2x{y^2} - x{y^2}} \right) + \left( {\frac{1}{2} - 1} \right)\]

\[ = 3{x^4} + x{y^2} - \frac{1}{2}\].

b) \(\left( {3{x^3} - {x^2}y + 2xy + 3} \right) - \left( {3{x^3} - 2{x^2}y - xy + 3} \right)\)

\( = 3{x^3} - {x^2}y + 2xy + 3 - 3{x^3} + 2{x^2}y + xy - 3\)

\( = \left( {3{x^3} - 3{x^3}} \right) + \left( { - {x^2}y + 2{x^2}y} \right) + \left( {2xy + xy} \right) + (3 - 3)\)

\( = {x^2}y + 3xy\).

c) \(\left( {{x^2} + 2xy - 3} \right)( - xy)\)

\[ = ( - xy)\,.\,{x^2} + ( - xy)\,.\,2xy + ( - xy)\,.\,( - 3)\]

\( =  - {x^3}y - 2{x^2}{y^2} + 3xy\).

d) \(\left( {15{x^5}{y^3} - 10{x^3}{y^2} + 20{x^4}{y^4}} \right):5{x^2}{y^2}\)

\( = \left( {15{x^5}{y^3}:5{x^2}{y^2}} \right) + \left( { - 10{x^3}{y^2}:5{x^2}{y^2}} \right) + \left( {20{x^4}{y^4}:5{x^2}{y^2}} \right)\)

\( = 3{x^3}y - 2x + 4{x^2}{y^2}\).