Simplifying (▲^2 - x▲)(x^3) Using Finite Differences

[irony of truth]

I am finding (▲^2 - x▲)(x^3).

I hope I am correct here...

(▲^2 - x▲)(x^3) = ▲^2(x^3) - x▲(x^3)
= ▲▲(x^3) - x( (x+1)^3 - x^3 )
= ▲( (x+1)^3 - x^3 ) - x( (x+1)^3 - x^3 )
= (x+2)^3 - (x + 1)^3 - (x+1)^3 + x^3 )
- x( (x+1)^3 - x^3 )

Is my solution correct?

 

[HallsofIvy]

No. Δ²x³ IS Δ(Δ x³)= Δ((x+1)³- x³)= Δ(3x²+ 3x+ 1)=
3(x+1)²+ 3(x+1)+ 1- (3x²+ 3x+ 1)= 6x+ 6.

 

[Hurkyl]

The work looks right, but you had really ought to simplify the expression. (Oh, and you need to stick an extra parenthesis in front)