Question

How do you multiply `(10x^2-13xy-3y^2)/(8x^2-10xy-3y^2)*(2y+8x)/(2x^2+2y^2)`?

Answer 1

`{40x^3-42x^2y - 25xy^2 - 3y^3 }/{8x^4 - 10x^3y + 5x^2y^2 - 10xy^3 - 3y^4)`

Explanation:

`{(10x^2 - 13xy - 3y^2)(2y+8x)}/{(8x^2 - 10xy - 3y^2)(2x^2 + 2y^2)}=`

`{2y(10x^2 - 13xy - 3y^2)+8x(10x^2 - 13xy - 3y^2)}/{2x^2(8x^2 - 10xy - 3y^2) + 2y^2(8x^2 - 10xy - 3y^2))`

`{2y10x^2 - 2y13xy - 2y3y^2 +8x10x^2 - 8x13xy - 8x3y^2}/{2x^2 8x^2 - 2x^2 10xy - 2x^2 3y^2 + 2y^2 8x^2 - 2y^2 10xy - 2y^2 3y^2)`

This can be simplified to:

`{10x^2y - 13xy^2 - 3y^3 +40x^3 - 52x^2y - 12xy^2}/{8x^4 - 10x^3y - 3x^2y^2 + 8x^2y^2 - 10xy^3 - 3y^4)`

and further:

`{40x^3-42x^2y - 25xy^2 - 3y^3 }/{8x^4 - 10x^3y + 5x^2y^2 - 10xy^3 - 3y^4)`

Answer 2

`color(magenta)(80x^3-84x^2y-50xy^2-6y^3)/(color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4)`

Explanation:

`(10x^2-13xy-3y^2)/(8x^2-10xy-3y^2)*(2y+8x)/(2x^2+2y^2)`

`color(white)(aaaaaaaaaaaaa)` `10x^2-13xy-3y^2`
`color(white)(aaaaaaaaaaa)` `xx underline(2y+8x)`
`color(white)(aaaaaaaaaaaaa)` `20x^2y-26xy^2-6y^3`
`color(white)(aaaaaaaaaaa)` `-104x^2y-24xy^2+0+80x^3`
`color(white)(aaaaaaaaaaaaa)` `overline(-84x^2y-50xy^2-6y^3+80x^3)`

`color(white)(aaaaaaaaaaaaa)` `color(magenta)(80x^3-84x^2y-50xy^2-6y^3`

`color(white)(aaaaaaaaaaaaa)` `8x^2-10xy-3y^2`
`color(white)(aaaaaaaaaaa)` `xx underline(2x^2+2y^2)`
`color(white)(aaaaaaaaaaaaa)` `16x^4-20x^3y-6x^2y^2`
`color(white)(aaaaaaaaaaaaaaaaaaaaaaaaaa)` `16x^2y^2-20xy^3-6y^4`
`color(white)(aaaaaaaaaaaaa)` `overline(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4)`

`color(white)(aaaaaaaaaaaaa)` `color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4`

`color(white)(aaaaaaaaaaaaa)` `color(magenta)((80x^3-84x^2y-50xy^2-6y^3)/color(blue)(16x^4-20x^3y+10x^2y^2-20xy^3-6y^4`