Question

How do you multiply `(2x^2 + y^2)(x - 2y)`?

Answer 1

By applying the distributive property several times you can get:
`(2x^2+y^2)(x-2y)`
`color(white)("XXXX")` `= 2x^3+xy^2-4x^2y-2y^3`

Explanation:

The distributive property tells us that
`color(white)("XXXX")` `(color(red)(a)+color(blue)(b))(color(green)(p)-color(orange)(q)) = (color(red)(a)+color(blue)(b))color(green)(p) - (color(red)(a)+color(blue)(b))color(orange)(q)`
and that
`color(white)("XXXX")` `(color(red)(a)+color(blue)(b))color(green)(p) = color(red)(a)color(green)(p)+color(blue)(b)color(green)(p)`

Therefore
`color(white)("XXXX")` `(color(red)(2x^2)+color(blue)(y^2))(color(green)(x)-color(orange)(2y))`

`color(white)("XXXX")` `(color(red)(2x^2)+color(blue)(y^2))color(green)(x) - (color(red)(2x^2)+color(blue)(y^2))color(orange)(2y)`

`color(white)("XXXX")` `=(color(red)(2x^2))(color(green)(x)) +(color(blue)(y^2))(color(green)(x)) - (color(red)(2x^2))(color(orange)(2y)) - (color(blue)(y^2))(color(orange)(2y))`

`color(white)("XXXX")` `=2x^3+xy^2 -4x^2y-2y^3`