Question

How do you evaluate `(a ^ { n + 2} - 2a ^ { n } + 3a ^ { n + 1} ) ( a ^ { n } + a ^ { n + 1} )`?

Answer 1

`-2a^(2n)+a^(2n+1)+4a^(2n+2)+a^(2n+3)`
OR
`a^(2n)*(-2+a+4a^2+a^3)`

Explanation:

Laws:
1) `x^a*x^b=x^(a+b)`
2) `b*x^a+c*x^a=(b+c)*x^(a)`

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`(a^(n+2)-2a^n+3a^(n+1))(a^n+a^(n+1))=`
`=a^(n+2+n)+a^(n+2+n+1)-2a^(n+n)-2a^(n+n+1)+3a^(n+n+1)+3a^(n+1+n+1)=`
`=a^(2n+2)+a^(2n+3)-2a^(2n)-2a^(2n+1)+3a^(2n+1)+3a^(2n+2)=`
`=4a^(2n+2)+a^(2n+3)-2a^(2n)+a^(2n+1)=`
`=-2a^(2n)+a^(2n+1)+4a^(2n+2)+a^(2n+3)=`

`=a^(2n)*(-2+a+4a^2+a^3)`