Find the coefficient of `x^4` in the expansion of `(2-x+3x^2)^6dot`

Explanation

We have,
`(2-x+3x^(2))^(6)=sum(6!)/(n_(1)!n_(2)!n_(3)!)2^(n_(1))(-x)^(n_(2))(3x^(2))^(n_(3))`
`=sum(6!)/(n_(1)!n_(2)!n_(3)!)2^(n_(1))(-1)^(n_(2))3^(n_(3))x^(n_(2)+2n_(3))`
where `n_(1),n_(2),n_(3)` are non-negative integers such that `n_(1)+n_(2)+n_(3)=6`
For coefficient of `x^(4)` in `(2-x+3x^(2))^(6)`, we must have `n_(2)=2n_(3)=4`
Since `n_(1),n_(2),n_(3)` are non-negative integers such that `n_(1)+n_(2)+n_(3)=6 and n_(2)+2n_(3)+4`,
which suggests that `n_(3) le 2`.
`{:(n_(1),n_(2),n_(3)),(4,0,2),(3,2,1),(2,4,0):}`
Hence the requried coefficient of `x^(4)` in the expansion of `(2-x+3x^(2))^(6)` is equal to
`(6!)/(4!0!2!)2^(4)(-1)^(0)3^(2)+(6!)/(3!2!1!)2^(3)(-1)^(2)3^(1)+(6!)/(2!4!0!)2^(2)(-1)^(4)3^(0)`
`=2160+1440+60=3660`