`a)`
`(2x^4y - x^3 + x^2 y^3) . 5x^2y`
` = 5x^2y . 2x^4y - 5x^2y . x^3 + 5x^2y . x^2 y^3`
` = 10 x^6 y^2 - 5 x^5 y + 5 x^4 y^4`
`b)`
`(10x^4 - 5x^3 + 3x^2) : 5x^2`
` = 10x^4 : 5x^2 - 5x^3 : 5x^2 + 3x^2 : 5x^2`
` = 2 x^2 - x + 3/5`
`c)`
`(x-4)(yx+1)`
` = x . yx + x . 1 - 4 . yx - 4 . 1`
` = x^2y + x - 4xy - 4`
`d)`
`(x^3 + 4x^2 - x - 4) : (x+1)`
` = [(x^3 + 3x^2 - 4x) + (x^2 + 3x - 4) ] : (x+1)`
`= [ x (x^2 + 3x -4) + (x^2 + 3x-4)] : (x+1)`
`= (x^2 + 3x - 4)(x+1) : (x+1)`
`= (x^2 - x + 4x - 4)(x+1) : (x+1)`
` = x^2 - x + 4x - 4`
`e)`
`(xy^2)^4 : (xy^2)^2`
` = ( xy ^2)^2`
` = x^2 y^4`